Mathematical Help

•Monomials and polynomials •Rational expressions -- definitions and operations •Absolute value (modulus) of numbers •Absolute value (modulus) of expressions •Percentages •Simplifying expressions •Powers •Roots •Domains •Multiplication		•Marking points using rectangular coordinates  •Functions •Equations •Inequalities •Simultaneous equations •Simultaneous inequalities •Inequalities linked with OR •Union (∪) and intersection (∩) of sets •Disjunction (∨,OR) and conjunction (∧,AND) of statements •Errors and messages


•Simultaneous equations -- basic concepts

•Substitution method

•Elimination method

•Graphical method


•Simultaneous inequalities -- basic concepts

•Simultaneous inequalities with one unknown

•Simultaneous linear inequalities with two unknowns

•Estimation -- advanced level


•Linear functions.

  ••Linear functions - definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Quadratic functions.

  ••Quadratic functions - definitions

  ••The graph of a quadratic function

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function -- basic level and advanced level


•Equations -- basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


•Inequalities -- basic concepts

•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Simultaneous linear inequalities with two unknowns

•Inequalities with one unknown -- advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Multiplication of rational expressions

•Division of rational expressions


•Monomials

•Polynomials

•Reduction of similar terms

•Multiplication and division of monomials

•Multiplication and division of polynomials by monomials

•Multiplication of polynomials

•Multiplication formulas

•Factorization of polynomials


Definition of a monomial
Monomials

A monomial is an algebraic expression which is either a number, a variable or the product of numbers and variables.

Examples


Definition of a polynomial
Polynomials

A polynomial is a sum of monomials.

Examples


Reduction of similar terms

Two monomials are similar if they contain the same variables as factors.

Reduction of similar terms means the addition (subtraction) of similar monomials.

To add (subtract) two similar terms we add (subtract) their numerical coefficients.

Examples


The product (quotient) of two numbers having the same sign is positive.

The product (quotient) of two numbers of opposite sign is negative.


The conditions which appear in the examples define the domain.

They result from the need for the divisor to be non-zero.


Multiplication and division of monomials

Revision of multiplication of numbers and powers

To multiply two monomials we multiply their numerical and variable factors using the laws of associativity and commutativity.

The result of multiplying monomials is always a monomial.

Examples

To divide two monomials we write them as the numerator and denominator of a fraction, and then cancel this expression.

The result of dividing monomials is not always a monomial.

In order for division to be possible the divisor must be non-zero.

Examples


Multiplication and division of polynomials by monomials

Multiplication and division of monomials

To multiply (divide) a polynomial by a monomial we multiply (divide) every term by the monomial.

Examples

The conditions which appear in the example 3 define the domain of the expression. They result from the need for the divisor to be non-zero.


Multiplication of polynomials

To multiply two polynomials we multiply each term in the first sum by each term in the second (multiplication of monomials). Next we carry out reduction of similar terms.

The result of multiplying polynomials is a polynomial.

Example


Multiplication formulas

The square of the sum of two quantities
The square of the difference of two quantities
The product of the sum of two quantities by their difference

The above formulas can be derived by multiplication of polynomials.


The square of the sum of two quantities
The square of the difference of two quantities
The product of the sum of two quantities by their difference


If a factor appears in every term of a sum, then we can take it outside a bracket.

1.	In this example the repeated factor is
2.	In this example the repeated factor is


Sometimes it is possible to group the terms of a sum in such a way that, after taking out a repeated factor in each group, a common factor for the entire sum becomes apparent.


We can often use multiplication formulas when factorizing.


Factorization of polynomials

Factorization of polynomials means expressing them as the product of two or more factors, each of which is a monomial or polynomial.

1. If a factor appears in every term of a sum, then we can take it outside a bracket.

     Examples

2. Sometimes it is possible to group the terms of a sum in such a way that, after taking out a repeated factor in each group, a common factor for the entire sum becomes apparent.

     Examples

3. We can often use multiplication formulas when factorizing.

     Examples

4. Putting the expression being factorized equal to zero, and solving the equation. Recommended for advanced students.


Factorization of polynomials means expressing them as the product of two or more factors, each of which is a monomial or polynomial.

Common methods of factorization are:

a) taking a factor outside a bracket

b) grouping terms

c) use of multiplication formulas

d) solution of a corresponding equation.

You can find examples, with commentary, in the chapter Factorization of polynomials.


•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

• Multiplication of rational expressions

• Division of rational expressions


Definition of a rational expression
The definition of a rational expression

See also

A rational expression is an expression which is the quotient of monomials or polynomials.

As in the case of common fractions, we can distinguish the numerator and denominator of a rational expression.

Examples of rational expressions

Note that the expression

	is not a rational expression since [image] is neither a monomial nor a polynomial.


•Definition of a rational expression

•Expansion of a rational expression

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Multiplication of rational expressions

•Division of rational expressions


The domain of the rational expression

is the set of non-zero real numbers, which can be written

y∈ℜ-{0} or y≠0.

In the program, the notation

y≠0

should be used.


The domain of the rational expression

is the set of pairs of numbers (a;b) such that

a≠1 and b≠-2x.


The domain of the rational expression

is the set of pairs of numbers (m;n) such that  m≠n.


The domain of the rational expression

is the set of triples of numbers (x;y;m) such that

x≠1 and y≠2x and m≠-6.

NB Sometimes the conditions which define the domain are separated by commas:

x≠1, y≠2x, m≠-6,

however this notation is not supported by the program 2xy.


To find the domain of the rational expression

we put the denominator equal to zero

When determining the domain we are interested in the points where the denominator is not equal to zero. We therefore negate the above solution:


Definition of the domain of a rational expression
The domain of a rational expression

See also

The domain of a rational expression is the set of all points at which the value of the denominator is non-zero.

Example 1.

Example 2.

Example 3.

Example 4.

Example 5.


The domain of a rational expression is the set of

all points at which the value of the denominator is non-zero.

You can find examples, with commentary, in the chapter

The domain of a rational expression.


•Definition of a rational expression

•The domain of a rational expression

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Multiplication of rational expressions

•Division of rational expressions


We expand the expression

by the sum m+1.

We obtain

Note that the domain has changed, as the equation m+1=0 has a solution.


We expand the expression

by the sum

We obtain

In this example we have not altered the domain, since


Method of expanding rational expressions
Expansion of rational expressions

See also   Revision of powers

To expand the rational expression

we multiply the numerator and denominator of the expression by a monomial or polynomial W.

NB When we expand a given expression we may change its domain. (This occurs when the equation W=0 has solutions.) In this case the initial expression and the expanded expression are equal on the new domain B≠0 and W≠0.

Example 1.

Example 2.

NB In the case of irrational expressions we proceed similarly. Only the domain changes.


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Multiplication of rational expressions

•Division of rational expressions


Method of canceling rational expressions
Cancellation of rational expressions

See also

To cancel (simplify) a rational expression we divide the numerator and denominator of the expression by factors which occur in both the numerator and denominator.

NB When we cancel an expression we may change its domain. The initial expression and the canceled expression are equal only on the domain of the initial expression.

When canceling it is helpful to factorize the numerator and denominator. Understanding of powers may also be useful.

Examples

NB In the case of irrational expressions we proceed similarly. Only the domain changes.


To cancel (simplify) a rational expression we divide the numerator and denominator of the expression by factors which occur in both the numerator and denominator.

NB When we cancel an expression we may change its domain. The initial expression and the canceled expression are equal only on the domain of the initial expression.

When canceling it is helpful to factorize the numerator and denominator.

Example

More examples can be found in the chapter Cancellation of rational expressions


To expand a rational expression

we multiply the numerator and denominator of the expression by a monomial or polynomial W.

NB When we expand a given expression we may change its domain. (This occurs when the equation W=0 has solutions.) In this case the initial expression and the expanded expression are equal on the new domain B≠0 and W≠0.

Example

You will find more information in the chapter Expansion of rational expressions.


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Multiplication of rational expressions

•Division of rational expressions


To add two rational expressions we put them over a common denominator, and then add their numerators.


Addition of rational expressions

See also

To add two rational expressions we put them over a common denominator, and then add their numerators (as we do with common fractions).

Remember! The line between the numerator and denominator takes the place of the division sign and brackets.

Examples

NB In the case of irrational expressions we proceed similarly. Only the domain is changed.


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Division of rational expressions


To multiply two rational expressions we multiply together their numerators, followed by their denominators.

In this example it is very important to state the domain, since cancellation means that the domain of the result is different from the domain of the initial expression.


To multiply two rational expressions we multiply together their numerators, followed by their denominators.

In this example it is very important to state the domain, since cancellation means that the domain of the result is different from the domain of the initial expression.


To multiply two rational expressions we multiply together their numerators, followed by their denominators.


To multiply a rational expression by a polynomial we multiply the numerator of the expression by the given polynomial.


Multiplication of rational expressions

See also

To multiply two rational expressions we multiply together their numerators, followed by their denominators.

If we are performing multiplication for purposes other than the expansion of the expression then it is good to cancel.

Example 1,    Example 2,    Example 3.

As usual, factorization of polynomials is useful when canceling a rational expression.

To multiply a rational expression by a monomial or polynomial we multiply the numerator of the expression by the given monomial or polynomial.

(The monomial or polynomial can be written in the form of a fraction with denominator 1.)

Example 4.

NB In the case of irrational expressions we proceed similarly. Only the domain changes.


To multiply two rational expressions

we multiply together their numerators,

followed by their denominators.

To multiply a rational expression by a monomial

or polynomial we multiply the numerator of the

expression by the given monomial or polynomial.

(The monomial or polynomial can be written in the

form of a fraction with denominator 1.)

You will find more information in the chapter

Multiplication of rational expressions.


The reciprocal of the expression A (A≠0) is the expression

Examples


Division of two rational expressions - examples.

1.

To determine the domain of the expression, we add to the conditions denoting the domain of the dividend (m-n≠0) the conditions necessary for the divisor to be well-defined (n≠0) and non-zero (m≠0).

2.

To determine the domain of the expression, we add to the conditions denoting the domain of the dividend (a+3≠0) the conditions necessary for the divisor to be well-defined (a+3≠0) and non-zero (a+1≠0).

3.

To determine the domain of the expression, we add to the conditions denoting the domain of the dividend (x≠0) the conditions necessary for the divisor to be well-defined ( [image]) and non-zero ( [image])


Division of a monomial by a rational expression.

The domain of the expression is determined by the conditions necessary for the divisor to be well-defined (x≠0) and non-zero (x+1≠0).

Division of a polynomial by a rational expression.

The domain of the expression is determined by the conditions necessary for the divisor to be well-defined (q≠0) and non-zero (p-1≠0).


Division of a rational expression by a polynomial.

To determine the domain it is necessary to find the conditions which are satisfied by the domain of the dividend (x+2≠0) and those which make the divisor non-zero (x-2≠0).


To divide, we multiply by the reciprocal.


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Putting rational expressions over a common denominator

•Addition of rational expressions

•Multiplication of rational expressions


Division of rational expressions

See also

To divide two expressions we multiply the dividend by the reciprocal of the divisor.

1. Division of two rational expressions.

To determine the domain of the expression, we add to the conditions which are satisfied by the domain of the dividend (B≠0) the conditions which make the divisor well-defined (D≠0) and non-zero (C≠0).   Examples

2. Division of a monomial (polynomial) by a rational expression.

In this case the domain of the expression is determined by the conditions which make the divisor well-defined (D≠0) and non-zero (C≠0).   Examples

3. Division of a rational expression by a monomial (polynomial).

To determine the domain it is necessary to find the conditions which are satisfied by the domain of the dividend (B≠0) and the conditions which make the divisor non-zero (C≠0).   Example

Note 1 If the expression contains several division signs (/) then after determining the domain it is best to change division operations into multiplication by the reciprocal.

Example

Note 2 If one of the expressions is irrational then we proceed similarly. Only the domain changes.


•Definition of a rational expression

•The domain of a rational expression

•Expansion of rational expressions

•Cancellation of rational expressions

•Addition of rational expressions

•Multiplication of rational expressions

•Division of rational expressions


If we have expressions with denominators

then to make the denominators equal we must expand the first expression by the factor x.


If we have expressions with denominators

then to make the denominators equal we must expand the first expression by the factor c.


If we have expressions with denominators

then the common denominator will be

We find this by taking the factors which appear in both denominators (in this case mn), and then multiplying by the remaining factors appearing in both denominators (in this case p,k). Once we have found the common denominator we have only to expand the fractions, the first expression by the factor p and the second by k.

We obtain


If we have expressions with denominators

then the common denominator will be

We find this by taking the factors which appear in both denominators (in this case u+1) and multiplying by the remaining factors from each denominator (in this case u+1,u-1). Now we have just to expand the fractions, the first expression by the factor u-1, and the second by u+1.

We obtain


If we have expressions with denominators

then we write them in such a way as to be able to see the common factors:

As usual we find the common factor by multiplying the factors which are common to both denominators; in this case

by the remaining factors; in this case

The common denominator is therefore equal to

We now expand the first fraction by the factor

and the second by

We obtain


Putting rational expressions over a common denominator

See also

To put two rational expressions over a common denominator we expand each of them so as to make the denominators of the two expressions equal. It is usually helpful to factorize the denominators.

Example 1.

Example 2.

Example 3.

Example 4.

Example 5.

NB In the case of irrational expressions we proceed similarly. Only the domain changes.


To put two rational expressions over a common denominator we expand each of them in such a way as to make both denominators equal.

It is usually helpful to factorize the denominators.

You will find examples, with commentary, in the chapter Putting rational expressions over a common denominator.


The quotient of expression A by expression B is the expression:


Absolute value (modulus) of numbers

The absolute value of zero or a positive number is the same number, e.g.

|5|=5

The absolute value of a negative number is obtained by changing the sign of that number, e.g.

|-10|=10

Note:

-10<5,   but   |-10|>|5|,      since      10>5


We draw a diagram of signs of expressions x and x-1.

We notice that

• for  x<0   - both expressions are negative

• for  0≤x<1   (x≥0 and x-1<0)

• for  x≥1   - both expressions are non-negative


We shall solve the equation

To do this we must consider three cases.

Case 1: x and x+1 are negative. |x|=-x |x-1|=-(x-1)	Case 2: x is non-negative and  x-1 is negative. |x|=x |x-1|=-(x-1)	Case 3: x and x+1 are non-negative. |x|=x |x-1|=x-1


Absolute value (modulus) of expressions

The absolute value of an expression a is equal to a when a is non-negative (a≥0), and is equal to -a when a is negative (a<0).

Theorems

Example of a transformation with absolute values

Example of an equation with absolute values

Example of an equation with absolute values

In the program, if two absolute values occurring next to each other are to be multiplied together, then we must put a multiplication sign between them, e.g. |x+1|∗|x| . The computer treats the notation |x+1||x| as incomplete and requests the insertion of an operation symbol. (To make it easier to see which of the vertical lines is treated by the computer as opening the absolute value notation and which as closing it, their shape will change for a short time.)

This is because the notation is sometimes ambiguous:

|a|b|c|

can be interpreted

as |a|∗b∗|c| or as |a∗|b|∗c|


We can use common fractions or decimals when performing the conversion. If a fraction has an exact decimal representation, then it does not matter whether we write it as a decimal or as a fraction.


If there is not an exact decimal representation, then it is better to use fractional notation, as this avoids approximation or the use of recurring decimals.


To convert a number to a percentage we multiply it by 100%.


Percentages

One percent denotes

Percentages are shown by the symbol %.

To convert a percentage to a number we multiply the number by

We can use common fractions or decimals when performing the conversion. If a fraction has an exact decimal representation, then it does not matter whether we write it as a decimal or as a fraction.

Examples

If there is not an exact decimal representation, then it is better to use fractional notation, as this avoids approximation or the use of recurring decimals.

Examples

To convert a number to a percentage we multiply it by 100%.

Examples


The brackets are unnecessary provided that the bracket is not raised to any power, and:

a) the bracket contains only one term
b) the bracket contains another bracket
c) the bracket contains the product of expressions
d) the bracket is not multiplied by another expression  NB If there is a minus sign in front of the bracket, then when the brackets are omitted it is necessary to change the sign of each expression inside the bracket.


1.We begin the transformation by performing the addition in the denominator of the expression.	2.We have put the expressions in the denominator of the main expression over a common denominator.	3.After removing the brackets.	4.After reduction of similar terms.
5.We factorize the denominator of the denominator, in order to cancel.	6.We have canceled the denominator.	7.We replace the main fraction line by a division sign (/).	8.Instead of dividing, we multiply by the reciprocal of the divisor.
9.We write the expression as a single fraction.	10.We write the expression in a more elegant form.		Domain


We change the division operation (represented by the fraction line) into multiplication by the reciprocal of the local denominator. Next we write the result obtained as a single fraction.

We can now deal with the main fraction (whose fraction line is level with the equality sign). Again we change division (represented by the fraction line) into multiplication by the reciprocal of the denominator, and we write the result of this multiplication as a single fraction.


Example of reducing layers.

	a) we change the main fraction line (division) into multiplication by the reciprocal of the denominator of the main fraction, b) we multiply the fractions,  c) we repeat the above operations.


Example of reducing layers.

	a) we change the main fraction line (division) into multiplication by the reciprocal of the main denominator, b) we change the main fraction line in the multi-layer fraction into multiplication by the reciprocal of its denominator, c) we perform the multiplication.

Another way of transforming the above expression:

	a) we change the main fraction line into multiplication by the reciprocal of the denominator  b) we carry out the multiplication c) we repeat the above operations


Example of reducing layers.

	I. We deal with the main denominator:  a) w change the main fraction line into multiplication by the reciprocal of the denominator,  b) we carry out the multiplication. II. We deal with the multi-layer fraction and change its main fraction line into multiplication by the reciprocal of the main denominator. III. We carry out the multiplication.

Another way to transform the same expression

	a) we change the main fraction line into multiplication by the reciprocal of the main denominator, b) we change the main fraction line in the multi-layer fraction into multiplication by the reciprocal of the main denominator, c) we carry out the multiplication.


Reducing the layers.

To reduce the layers we multiply the numerator by the reciprocal of the denominator.

	If any of the expressions A,B,C,D itself contains several layers then we simplify the form of the expression in stages, using operations analogous to those illustrated here. Example 1. Example 2. Example 3.

If the numerator or denominator of any layer is the sum of rational expressions, then we put it over a common denominator, write it as a single fraction and try to cancel. Next we replace the division operation (represented by the fraction line) with multiplication by the reciprocal of the denominator.

Remember that the fraction line takes the place of brackets and a division sign.

If our expression does not contain irrational roots then we can write it in the form of a rational expression.

Example


When simplifying expressions with negatives it is useful to remember the following facts:

a) when we multiply (divide) two numbers with the same sign we obtain a positive number

b) when we multiply (divide) two numbers with opposite signs we obtain a negative number

and these formulas:


Negative whole-number powers are defined as follows

Negative whole-number roots are defined as follows


What do we understand by simplifying expressions?
Simplifying expressions

Simplifying an expression involves:

• calculating all exponents

• calculating powers where the exponent or the base is equal to 0 or 1

• performing operations involving zero

• performing operations involving one

• writing fractions without redundant zeros and ones

• carrying out reduction of similar terms

• replacing powers with negative exponents by powers with positive exponents

• removing unnecessary brackets

• reducing multi-layer fractions

• simplifying expressions with negatives

• performing operations involving numbers and monomials.


Multiplication of powers with the same base


Dividing powers with the same base


Multiplying powers with the same exponent


Dividing powers with the same exponent


The power of a power


For [image] we have:
	a power with a negative whole-number exponent
	multiplication of powers with the same base
	division of powers with the same base
	multiplication of powers with the same exponent
	division of powers with the same exponent
	the power of a power


Examples of calculations involving powers.

1.

We used the theorem concerning multiplication of powers with the same base,

multiplication formulas and multiplication of polynomials.

2.

In simplifying the denominator we made use of the theorem concerning division of powers with the same base.

3.

We used, in order, the definition of powers with negative exponent  and the theorems concerning division of powers with the same exponent,  multiplication of powers with the same exponent,  raising a power to another power,  and  division of powers with the same base.


Definitions and theorems concerning powers
Powers

Definition of powers with natural-number exponents a is called the base n is called the exponent
Definition of powers with negative whole-number exponents
Basic theorems concerning raising expressions to the power of a natural number.

	multiplication of powers with the same base
	division of powers with the same base
	multiplication of powers with the same exponent
	division of powers with the same exponent
	the power of a power
NB Analogous results hold for negative whole-number exponents, the only difference being in the domain (in accordance with the above definition).

Examples


Theorem on the root of a power.


Theorem on the power of a root


•Definitions

•Theorems

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Domains


Examples of calculations involving roots

See also    Mathematical symbols

1. We wish to simplify as far as possible the expression

From the theorem on the root of a power  and the definition of absolute values, we have:

2. We wish to simplify the expression

The domain of the expression is given by the inequality [image].

We note that, as long as this condition is fulfilled, the following equality holds:

Hence

3. We wish to change the root of a product [image] into the product of roots.

The domain of our expression is given by the inequality [image].

The solution of the above inequality can be expressed as follows:

We have four cases:

I. a>0 AND b>0	II. a<0 AND b<0	III. a=0	IV. b=0
	a<0 ⇒ -a>0
Summing up:


Mathematical symbols:

∨ denotes OR

∧ denotes AND

⇔ denotes equivalence

                         if and only if

⇒ denotes implication

                         if ... then ...


•Definitions

•Theorems

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Addition and subtraction of roots

•Harder examples


Rationalization of the denominator.
Rationalization of the denominator

See also

If the denominator of a fraction contains an irrational root, them we say we have an irrationality in the denominator.

Rationalization of the denominator involves expanding the fractions in such a way as to eliminate the irrationality from the expression in the denominator.

Examples.

1.
2.
3.

Note. Sometimes rationalization of the denominator can be very difficult, for example when there are three roots there.


Definitions and formulas involving roots
 Roots

•Definitions

•Theorems

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Harder examples


•Definitions

•Theorems

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Rationalization of the denominator

•Harder examples


Description of method of adding and subtraction of roots.
Addition and subtraction of roots

See also

Two roots can be added if both the indexes and the values under the root sign are equal.

The expressions below cannot be added.

Sometimes the fact that addition is possible becomes clear only after we have made certain transformations.

In some cases, although addition is not possible, a factor can be taken outside the brackets. This can sometimes be done in several ways, and it is hard to claim that any one way is better than the others.


For odd p or a≥0 the equality follows easily from the theorem on the root of a power.

On the other hand, for even p and a<0 we have


•Definitions

•Theorems

•Taking a factor outside the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Harder examples


Bringing a factor under the root sign.
Bringing a factor under the root sign

See also

We should learn the following theorem on bringing a factor under the root sign.

For p∈N and p>0 we have

 Proof.

Examples

1.
2.
3.
4.


•Definitions

•Theorems

•Bringing a factor under the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Harder examples


Taking a factor outside the root sign.
Taking a factor outside the root sign

See also      Theorems

From the theorems on roots of powers and the multiplication of roots of the same degree, we have

Examples


For p∈N and p>0 we have:

1.		Theorem on multiplication of roots of the same degree.
2.		Theorem on division of roots of the same degree.
3.		Theorem on the root of a power.
4.		Theorem on the power of a root.


•Definitions

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Harder examples


Theorems connected with roots
Roots - theorems

See also

For p∈N and p>0 we have:

1.		Theorem on multiplication of roots of the same degree.
2.		Theorem on division of roots of the same degree.
3.		Theorem on the root of a power.
4.		Theorem on the power of a root.
Examples

furthermore, for m,n,q∈N and m,n,q>0

5.		Theorem on bringing a power under the root sign.
6.		Theorem on reducing the degree of a root (1)
7.		Theorem on reducing the degree of a root (2)
8.		Theorem on multiplication of roots of different index.
For roots of negative whole-number index analogous theorems hold, the only difference being in the domain.


•Theorems

•Bringing a factor under the root sign

•Taking a factor outside the root sign

•Rationalization of the denominator

•Addition and subtraction of roots

•Harder examples


Definitions connected with roots
Roots - definitions

See also

•The p-th root (p∈N, p>0) of a non-negative number a (a≥0) is defined as

the non-negative number b (b≥0) such that [image].

•Roots of negative numbers are defined only in the case of roots of odd index.

•Roots whose indices are negative whole numbers are defined as follows:

•Roots having an infinite, non-recurring decimal expansion (which cannot be written either as a whole number or as a fraction) are called irrational. Examples are:

When we write

we are referring to the square root

.

Examples

Remember that:

1. There are no roots of index 0.

2. Negative numbers do not have even-numbered roots.


•The p-th root (p∈N, p>0) of a non-negative number a (a≥0)

is defined as the non-negative number b (b≥0) such that [image].

•Roots of negative numbers are defined only in the case of roots of odd index.

•Roots whose indices are negative whole numbers are defined as follows:

•Roots having an infinite, non-recurring decimal expansion (which cannot be written either as a whole number or as a fraction) are called irrational.


We shall solve the inequality

We solve it on the assumption that [image] this having been

determined as the condition for the domain.

An inequality with the sign ≠ we solve like an equation

negating the final answer.


The inequality

is solved under the following assumptions:

these having been determined as the

conditions for the domain.


To find the domain of the expression

 we construct a set of simultaneous inequalities.

		← even roots are defined only for non-negative values
		← we cannot divide by zero (the fraction line denotes division)
(∗)		← we cannot divide by zero. To solve the last inequality we find its domain
			← we cannot divide by zero
		and then the solution
			← from solution of the inequality (∗)

We now solve the set of simultaneous inequalities:

	1.	2.	3. domain written in simplest form


To find the domain of the expression

we construct a set of simultaneous inequalities:

	← even roots are defined only for non-negative expressions
	← roots with negative index are defined only for non-zero expressions
It can easily be seen that this set of inequalities can be written:

	(∗)
The conditions for the above fraction to exist are

	←only non-zero expressions can be raised to the power zero,
	← we cannot divide by zero
and the solution of the inequality (∗):

We solve the set of inequalities obtained

	1.	2. domain written in simplest form


Description of method of finding the domains of various expressions.
Domains

The domain of an expression is the set of points where that expression makes sense.

When determining the domain, it is useful to remember the following facts:

• we cannot divide by zero,

• only non-zero expressions can be raised to the power zero,

• only non-zero expressions can be raised to a negative power,

• roots of index zero do not exist,

• negative expressions do not have roots of even index,

• expressions whose value is zero do not have roots with negative index.

NB Sometimes the conditions defining the domain are written separated by commas: x≠1, y≠2x, m≠-6, but this notation is not recognized by the program 2xy. In this program the correct notation requires the use of connectives instead of commas: x≠1 AND y≠2x AND m≠-6 or the writing of the conditions one under the other.

Example 1.

Example 2.


•Multiplication and division of a polynomial by a monomial

•Multiplication of polynomials

•Multiplication of rational expressions

•Division of rational expressions

•Roots

•Domains


Description of methods for the multiplication of various expressions.
Multiplication

See also

In this chapter we present basic formulas for multiplication.

The calculation of the domain is left to the user.

For p,q,m,n∈N-{0} we have

A simple corollary of the above formulas is the method of multiplying a bracket by a root.


The number line is a line

•with a distinguished starting point (usually 0),

•with a unit of distance

•and with a defined positive direction.

NB Every real number corresponds to a point on the number line.


The rectangular system of plane coordinates is based on two perpendicular axes, each of which is a number line.

NB

•The axes do not have to intersect at the point (0;0).

•The units of distance used on the two axes may be different.


•The graph of a linear function

•Graphs of linear functions from coefficients

•The graph of a quadratic function

•The graph of a linear quotient function -- basic level and advanced level


Method of marking points using rectangular coordinates.
Marking points using rectangular coordinates

See also

To mark the point with coordinates (p;q)  in the rectangular system of coordinates  •we find the point p on the horizontal axis and draw a line through this point perpendicular to the horizontal axis, •then we find the point q on the vertical axis and draw a line through it perpendicular to the vertical axis •the point (p;q) lies at the intersection of the two lines we have drawn.
If we are given a point in the rectangular coordinate system and we want to find its coordinates:  •we draw two lines through the point, perpendicular to the horizontal and vertical axes respectively. •The point of intersection of the first line with the horizontal axis gives the first coordinate of our given point, •and the point of intersection of the second line with the vertical axis gives the second coordinate.
Notice that  •points lying on the horizontal axis have their second coordinate equal to 0, ie. they are of the form (a;0) •points lying on the vertical axis have their first coordinate equal to 0, ie. they are of the form (0;a).


Increasing linear function. y=ax+b, a>0


Decreasing linear function. y=ax+b, a<0


Constant function. y=b. (y=ax+b, a=0)


Graphs of functions with equal coefficients of gradient.


For the linear function y=ax+b, if the same

units are used on each axis, we have a=tanα


The greater the coefficient of gradient, the steeper is the slope.


•Linear functions.

  ••Linear functions - definitions

  ••The graph of a linear function

•Quadratic functions.

  ••Quadratic functions - definitions

  ••The graph of a quadratic function

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function: basic level and advanced level


Description of graphs of linear functions from their coefficients.
Graphs of linear functions from coefficients

See also

•When the coefficient of gradient is positive (a>0) the linear function y=ax+b is increasing.

   Example

•When the coefficient of gradient is negative (a<0) the linear function y=ax+b is decreasing.

   Example

•The graph of the constant function y=b, (y=0•x+b) is a line parallel to the x axis which meets the y axis at the point b.

   Example

•If we have linear functions which differ only in their constant terms (their coefficients of gradient are equal) then their graphs are parallel to each other.

   Example

•If the units of distance on both axes are equal, then for the linear function y=ax+b

the coefficient a is equal to the tangent of the angle made by the graph of the function with the x axis.

a=tanα

   Example

•When the coefficient of gradient is positive, the greater the value of the coefficient, the steeper is the graph of the function.

   Example


•Linear functions.

  ••Linear functions - definitions

  ••Graphs of linear functions from coefficients

•Quadratic functions .

  ••Quadratic functions -definitions

  ••The graph of a quadratic function

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••Graphs of linear quotient functions --basic level and advanced level

•Marking points using rectangular coordinates


Graphs of linear functions and how to draw them.
The graph of a linear function

See also

The graph of a linear function of the form y=ax+b is a straight line.

To determine the line uniquely we need to know only two points.

We shall draw the graph of the function y=x+1.

We can take any values as arguments (x), eg. 2 and 4. We calculate the value of the function at these points,

for x=2 we have y=2+1=3, for x=4 we have y=4+1=5.

We place the arguments and corresponding values of the function in a table.

	The graph of our function passes through the points (2;3) and (4;5).  We mark them in the coordinate system and draw a line through them.
If a≠0 and b≠0 then we can choose to calculate the points where the graph intersects the two coordinate axes. Highly recommended!

	•The calculation of the value for x=0 is very easy. •The find the x coordinate of the second point we must solve a simple equation  0=x+1. This gives x=-1.


•Linear functions.

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Quadratic functions .

  ••Quadratic functions -definitions

  ••The graph of a quadratic function

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function -- basic level and advanced level


Description of linear functions.
Linear functions

See also

A linear function is a function of the form:

	x→ax+b f:x→ax+b    we call the function f f(x)=ax+b    we call the function f y=ax+b    this notation is very convenient when drawing graphs
a,b denote constants.

a is called the coefficient of gradient, b the constant term.

A constant function is a linear function whose coefficient of gradient is zero (a=0). It has the form

y=b.

The domain of a linear function is the set of all real numbers.

Examples

1.	2.	3.
y=2x	y=3x-1	y=-2  (constant function)
f(x)=2x	g(x)=3x-1	h(x)=-2
a=2, b=0	a=3, b=-1	a=0, b=-2

The graph of a linear function is a straight line.

Linear functions have parallel graphs if and only if their coefficients of gradient are equal.


•Quadratic functions.

  ••The graph of a quadratic function

•Linear functions.

  ••Linear functions-definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function -- basic level and advanced level


Description of quadratic functions.
Quadratic functions

See also

A quadratic function (quadratic trinomial) is a function of the form

a,b,c are fixed constants, called the coefficients of the quadratic function.

The domain of a quadratic function is the set of all real numbers.

The discriminant of a quadratic trinomial is the number

If the discriminant of a quadratic trinomial is non-negative (Δ≥0) then it is possible to write the function in the form of a product of linear factors

,

where

.

We can also write a quadratic trinomial in canonical form


A zero point of a function is a point at which the value of the function is equal to zero.


The graph of the quadratic function

depends on a and Δ.

Δ>0 2 zero points,
Δ=0 1 zero point,        (repeated root)
Δ<0  no zero points


We shall draw the graph of the function

We start by filling in a table.

A very important point is the vertex.

The remaining points can be chosen arbitrarily, so they can

be zero points of the function (y=0) or points where the graph

intersects the y axis (x=0).

	Points from the table are marked on the coordinate diagram and joined up appropriately.


•Quadratic functions.

  ••Quadratic functions -definitions

•Linear functions.

  ••Linear functions-definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function -- basic level and advanced level


Description of graphs of quadratic functions.
The graph of a quadratic function

See also

The graph of the quadratic function

is a parabola whose vertex is the point:

Example

From here on we shall assume a system of rectangular coordinates based on axes which intersect at the point (0;0) and are oriented as in the diagram.

•When a is positive (a>0) the branches of the parabola point upwards.

•When a is negative (a<0) the branches of the parabola point downwards.

•The number of zero points a quadratic function has depends on the sign of the discriminant of the quadratic trinomial:

   for Δ>0 the function has 2 zero points,

   for Δ=0 the function has 1 zero point (a repeated root),

   for Δ<0 the function has no zero points.

Illustration


It is easy to see that the graphs of the functions

are symmetric about the x axis.


It is easy to see that the graphs of the functions

are symmetric about the x axis.


A zero point of a function is a point at which the value of the function is equal to zero.

A zero point of a linear rational function is a point where the numerator is equal to zero.

Some linear rational functions do not have zero points, eg.


The line with equation x=xo is called a vertical asymptote of the curve y=f(x) if the function tends to plus or minus infinity at that point (from the left or from the right), ie.

The line with equation y=mx+n is called a diagonal asymptote of the curve y=f(x) if both

or

A diagonal asymptote is horizontal if m=0.


The linear rational function

has the asymptotes:

•vertical	and	•horizontal
At this point the function is undefined.		The function never takes this value.


Asymptotes of a linear rational function
Asymptotes of a linear rational function.

The graph of a linear quotient function -- basic level and advanced level

The linear rational function

has the asymptotes:

•vertical

  At this point the function is undefined.

•horizontal

  The function never takes this value.


•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function --basic level

•Linear functions.

  ••Linear functions - definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Quadratic functions .

  ••Quadratic functions - definitions

  ••The graph of a quadratic function


Drawing the graph of a linear rational function, advanced level.
The graph of a linear rational function, advanced level

See also     Definition of asymptotes

We shall draw the graph of the function

We start by completing the table. In the table on the screen, in the cell marked:
•x≠... we write the vertical asymptote, and in the cell marked

•y≠... we write the horizontal asymptote,

•in the remaining cells we write various values of x and calculate the corresponding values of the function at these points. If they exist, it is good to include the zero point and point of intersection with the y axis (x=0) in the table.

Next we mark the points on the coordinate diagram and join them up appropriately. In the program, in order to mark the asymptotes  x≠p, y≠q  we mark the point (p;q) and the computer draws the lines on the screen automatically. The graph of the function


•Linear quotient functions. Hyperbolas.

  ••Linear quotient functions - definitions

  ••The graph of a linear quotient function -- advanced level

•Linear functions.

  ••Linear functions - definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Quadratic functions.

  ••Quadratic functions - definitions

  ••The graph of a quadratic function


Method of drawing a graph of a linear rational function, basic level.
The graph of a linear rational function, basic level

See also

We shall draw the graph of the function

We start by completing the table. In the table, in the cell marked •x≠... we write the condition defining the domain, and in the cell
•y≠... we write the value which y can never take,

•in the remaining cells we write various values of x and calculate the corresponding values of the function at these points. Example

Next we mark the points on the coordinate diagram and join them up appropriately. In the program, to indicate the conditions x≠0, y≠0 we mark the point (0;0) and the computer draws the asymptotes on the screen automatically. The graph of the function


•Linear quotient functions. Hyperbolas.

  •Graphs of linear quotient functions -- basic level and advanced level

•Linear functions.

  ••Linear functions - definitions

  ••The graph of a linear function

  ••Graphs of linear functions from coefficients

•Quadratic functions .

  ••Quadratic functions - definitions

  ••The graph of a quadratic function


Description of linear rational functions.
Linear quotient functions - definitions

See also

A linear rational function is a function which is the quotient of two linear functions.

a,b,c,d denote constants.

The domain of a linear rational function is the set of real numbers for which the denominator is non-zero.

The range of a linear rational function is the set of all real numbers excluding the value a/c.

The graph of a linear rational function is a hyperbola.

••A special case of a linear rational function is the function

with domain given by x≠0 and range y≠0.


The equation

	has no solutions since the square of any number is non-negative
	has one solution  x=4
	has two solutions x=2 OR x=-2
	has infinitely many solutions, as any number satisfies this equation


•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


Basic concepts connected with equations.
Equations -- basic concepts

See also

An equation consists of two algebraic expressions, at least one of which contains an unknown, joined by an = sign.

A solution of an equation in n unknowns consists of n numbers such that, on substituting them for the unknowns, we obtain equality (ie. the left-hand side of the equation is made equal to the right-hand side).

To solve an equation means to find all of its solutions.

An equation may have no solutions, one solution, two solutions, ... , or it may have infinitely many solutions. Examples

If an equation has no solutions then we call it a contradiction.

If an equation is satisfied no matter which values the unknowns appearing in it take then we call it an identity.

Equations are called equivalent if they have the same domain and the same set of solutions.

Theorms on equivalence of equations

•If we carry out reduction of similar terms and perform operations appearing on one or both sides of the equation then the old and new equations are equivalent. Example

•If we add (subtract) the same term to (from) each side of the equation, then the old and new equations are equivalent. Example

•If we multiply or divide both sides of the equation by a non-zero number, then the old and new equations are equivalent. Example


Reduction of terms and performing operations appearing

on either side of the equation results in an equivalent equation.


We can add (subtract) the same term to (from) both sides

of the equation, and we still obtain an equivalent equation.


We can multiply (divide) both sides by the same non-zero

number, and we still obtain an equivalent equation.


A first-degree equation in two unknowns.

   ax+by+c=0,   where  x,y are the unknowns,  a,b,c are constants


We shall solve the simple equation

2x+3=4x-1.

When we solve an equation we group the terms containing the unknown on one side of the equation, and the remaining terms on the other. We group the terms containing x on the left-hand side. To do this we add -4x to each side of the equation, causing the term containing x to disappear from the right-hand side, since

4x+(-4x)=4x-4x=0.

We have

  2x+3=4x-1   |-4x

  2x+3-4x=4x-4x-1

  2x+3-4x=-1

We usually simplify this reasoning by saying that we move the given term from one side of the equation to the other with change of sign.

  2x+3=4x-1

  2x+3-4x=-1

After simplification we obtain  -2x+3=-1

Now we must get rid of the 3 on the left-hand side of the equation.

This can be done by adding -3 to both sides, since

3+(-3)=3-3=0.

Hence

-2x+3=-1   |-3 -2x+3-3=-1-3	For simplicity we can say that we have moved 3 to the other side of the equation with change of sign.	-2x+3=-1 -2x=-1-3
After simplification we obtain  -2x=-4

We now divide both sides by the coefficient which appears next to the unknown

-2x=-4   |/(-2)

and we obtain the solution:  x=2


•Equations - basic concepts

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


We shall solve the equation

		We eliminate the brackets on both sides of the equation.
		We carry out reduction of similar terms.
		We want the unknown x to appear only on one side (eg. on the left-hand side) of the equation. To achieve this we add -x to both sides, since x+(-x)=x-x=0. For simplicity we can say that we have moved the x to the other side of the equation with change of sign.
		We combine similar terms.
		Numbers are collected on the opposite side of the equation from the unknown terms. To eliminate the 3 from the left-hand side we add -3 to both sides, since 3+(-3)=3-3=0. For simplicity we say that we have moved the 3 to the other side of the equation with change of sign.
		We combine similar terms.
		To find x it is enough to divide both sides of the equation by the coefficient which appears next to the unknown.


Definition and solution of linear equations.
Linear equations

See also

A linear equation is one which can be transformed to an equivalent equation of the of the form

                                     ax+b=0,

where a,b denote constants and x is the unknown.

A linear equation is also called a first-degree equation (the unknown appears to the power one).

If we have n unknowns, all to the power one, then the equation is called a first-degree equation in n unknowns. Example 1.

Example 2.

Example 3.


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


Definition and solution of product equations.
Product equations ab=0.

See also

Product equations are equations of the form

ab=0

The equation holds when either of the factors is equal to zero.

Hence

a=0  OR  b=0

In the case where there are three factors, we have

abc=0

a=0  OR  b=0  OR c=0

When there are more factors the method of solution is analogous to the above.


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots

•Quadratic functions -definitions

•Graphs of quadratic functions


Definition and solution of quadratic equations.
Quadratic equations

See also

A quadratic equation is an equation of the form:

a,b,c constant.

The number of solutions of a quadratic equation depends on the sign of the discriminant of the quadratic trinomial:

for Δ>0 the quadratic equation has two different roots

for Δ=0 the quadratic equation has a repeated root

for Δ<0 the the quadratic equation has no solutions.

Example (advanced)


We shall solve the equation

in terms of the variable x.

We assume that the program is set for STRICT DOMAIN CHECKING.

We begin by finding the discriminant of the quadratic function.

To find the roots of the equation we use the square root of Δ, therefore in order to make sure we have the correct domain we must consider two cases: Δ≥0 and Δ<0.

In the program we open a new window (SHIFT+Y), fill in the appropriate conditions for the domain and press ENTER.

We now consider two cases.

Case 1	We click on contradiction on the on-screen keyboard. After checking, the program will automatically reduce this case to an icon. NB If the program is not set for  STRICT DOMAIN CHECKING  then consideration of this case is not necessary.
Case 2

	Here we have two solutions
We now click the answer button.


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


Definition and solution of biquadratic equations.
Biquadratic equations

See also

A biquadratic equation is an equation of the fourth degree of the form:

By means of the substitution

we can reduce a biquadratic equation to a quadratic equation.

Example


We shall solve the equation

Being a symmetric equation of odd degree, it has the root

x=-1.

We find the reduced equation and solve it.

Hence our equation has three roots


We shall solve the equation

We start by dividing both sides of the equation by

.

Notice that, although by dividing both sides by

we change the domain of the equation, we do not change the set of solutions, as x=0 is not a root of the equation.

After transforming the left-hand side of the equation and making the substitution

we obtain a quadratic equation

having two roots

Returning to the original variable

Hence our symmetric equation has no roots.


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


Definition and solution of symmetric equations.
Symmetric equations

See also

A third-degree equation is called symmetric when it has the form:

A fourth-degree equation is called symmetric when it has the form:

In general a symmetric equation is an equation of the form:

•A symmetric equation of odd degree always has the root

x=-1.

We therefore start by dividing the equation by x+1, and then look for roots of the reduced equation.

Example 1.

•When solving a symmetric equation of even degree we use the substitution

Example2.


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Use of the factor theorem in solving equations

•Equations involving roots


Definition and solution of equations of the type a^n=b.
Equations of the form [image].

See also

The solution of the equation

 ,  n∈N and n>0

is dependent on whether n is odd or even.

•for odd n

•for even n and b≥0

•for even n and b<0

there are no solutions

Example


•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Use of the factor theorem in solving equations

•Equations involving roots


The factor theorem.
The factor theorem

See also

The factor theorem

An equation of the n-th degree of the form:

W(x)=0

has the root

x=a

if and only if the polynomial W(x) is divisible by x-a, ie. if there exists a polynomial W1(x) of degree (n-1) such that:

W(x)=(x-a)W1(x).

It follows from the factor theorem that if we know a root of an equation of degree n

W(x)=0,

then we can reduce the solving of this equation to the solving of an equation of degree (n-1)

W1(x)=0.

The equation W1(x)=0 is called a reduced equation.


Useful theorems for solving equations.
Theorems for guessing roots of equations.

Applications of the factor theorem

Theorem 1.

If the integer p is a root of the equation

with whole-number coefficients

then p is a divisor of the constant term

Theorem 2.

If a rational number

	(p,q co-prime)
is a root of the equation

with whole-number coefficients

then p is a divisor of the constant term

and q is a divisor of the coefficient


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations involving roots

•Theorems for guessing solutions


Applications of the factor theorem.
Applications of the factor theorem to the solution of equations

See also

Example 1.

Example 2.


We shall solve the equation

.

The theorem for guessing solutions tells us that we should look for roots among the numbers 1,-1,2,-2,3,-3,6,-6, these being the factors of the constant term (-6).

Let

We find out which of the factors satisfy the equation W(x)=0.

We have:

W(1)=1+2-5-6≠0

W(-1)=-1+2+5-6=0

So -1 is a root of our equation.

On the basis of the factor theorem we can now look for the roots of the reduced equation, which in this case is the quadratic equation

whose roots are 2 and -3.

Note

It is clear that W(2)=0 and W(-3)=0. If we had checked that before finding the reduced equation, then we would already have solved the equation W(x)=0. This follows from the theorem which says that the number of solutions of an equation cannot exceed its degree. The degree of our initial equation is 3, and its solutions are

x=-1 or x=2 or x=-3


We shall solve the equation

We look for rational roots

The fractions we look for depend on the theorem on guessing solutions.

The factors of the constant term are 1,-1,3,-3 (these give p).

The factors of the coefficient of the highest power of  x  are 1,-1,2,-2 (these give q).

We now construct all possible fractions of the form

cancel them and delete those which are repeated. Hence

We notice that, of the above numbers, only 0.5 satisfies the initial equation.

By the factor theorem we can now look for roots of the reduced equation, which in this case is the quadratic equation

whose roots are

Our equation therefore has three roots


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Use of the factor theorem in solving equations

•Equations involving roots


Quotient equation.
Quotient equations.

See also

When solving quotient equations it is useful to remember the equivalence

Example

To solve the equation

we use the above equivalence

Hence


•Equations - basic concepts

•Linear equations

•Equations in many variables

•Product equations ab=0

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Use of the factor theorem in solving equations


Methods of solution of equations involving roots.
Equations involving roots

See also

•The solution of an equation of the form:

	where n∈N and n>0,
depends whether the index of the root is odd or even.

	For odd n  it is enough to raise both sides to the power n.
	For even n  •to the domain we add the condition for the root to be well-defined •we raise both sides to the power n, •to the equation we add the condition that b be non-negative (so as not to bring in extra roots).
Example 1

Example 2

•Some equations involving roots can be solved by substitution.

Example 3


We shall solve the equation

We make the substitution

We obtain an easily solved quadratic equation

We return to the original variable and solve

a simple equation with roots

The solutions of the equation are therefore x=4 or x=9.


We shall solve the equation

by expressing y as a function of  x.

To do this: 1) we add the condition denoting the domain,  x-y≥0, 2) we square both sides and find y, 3) we add to the equation the condition which guarantees that when we square both sides we do not bring in any extra roots, x≥0.

We now check that the solution satisfies the conditions defining the domain. We substitute the value of y into the condition. After simple transformations we obtain the inequality

This is satisfied by any value of x.

We can therefore remove the domain condition.

In the program we now click the answer button.


We shall solve the equation

To do this

1) we move one of the roots to the other side of the equation,

2) we add the conditions denoting the domain,  z-1≥0 ∧ z≥0, 3) we square both sides, 4) we add to the equation the condition which guarantees that when we square both sides we do not bring in any extra roots.
After some transformations we obtain:
Now we square both sides again to find z.
We now check that the solution satisfies all the conditions.We substitute the value of z into the inequalities.
We can therefore remove them.
In the program we now click the answer button.


In the program, when we want to open a new case we use the keys SHIFT+Y


•Inequalities - basic concepts

•Intervals and inequality signs

•Linear inequalities

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots

•Simultaneous inequalities - basic concepts

•Inequalities linked with OR


Product inequalities.
Product inequalities of the type ab>0

See also

Inequality	Solution
ab>0 ⇔	a>0  AND  b>0    OR    a<0  AND  b<0
ab<0 ⇔	a>0  AND  b<0    OR    a<0  AND  b>0
ab≥0 ⇔	ab=0    OR   ab>0
ab≤0 ⇔	ab=0    OR   ab<0
ab≠0 ⇔	a≠0  AND  b≠0

When there are three variables we have:

abc>0  ⇔  (ab)c>0  ⇔

⇔ ab>0  AND  c>0  OR  ab<0  AND  c<0  ⇔

⇔ (a>0  AND  b>0  OR  a<0  AND  b<0)  AND  c>0  OR  (a>0  AND  b<0  OR  a<0  AND  b>0)  AND  c<0

First we treated two variables as one, and then broke down the remaining product.

We use an analogous method when we have a larger number of variables.

NB If a product inequality contains only one variable then it is usually easier to sketch a graph and read off the solution from the graph.


Multiple inequalities.
Multiple inequalities

Sometimes we write inequalities using several >,≥ symbols or <,≤.symbols.

In an inequality so written can appear either only the signs >,≥,= or only the signs <,≤,=.

Using the connective AND we can easily convert a multiple inequality into single inequalities.

   x<y≤z   ⇔   x<y   AND   y≤z

   w>x>y≥z   ⇔   w>x   AND   x>y   AND   y≥z

   x=y≥z   ⇔   x=y   AND   y≥z


•Inequalities - basic concepts

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots


Intervals and inequality signs.

See also

Open intervals (,) correspond to strict inequalities >,<, and closed intervals [,] to weak inequalities ≥,≤. When infinity is involved (∞) we always use open intervals.

In the program we use semi-colons (;) between the expressions in interval notation.

x∈(5;+∞) ⇔ x>5

x∈[5;+∞) ⇔ x≥5

x∈(-∞;3) ⇔ x<3

x∈(-∞;3] ⇔ x≤3

x∈(1;3) ⇔ x>1  AND  x<3

x∈[1;3] ⇔ x≥1  AND  x≤3

x∈(2;4] ⇔ x>2  AND  x≤4

x∈[2;4) ⇔ x≥2  AND  x<4

In the case of an open interval and strict inequality, the end-point is not a solution of the inequality.

x∈(5;+∞) ⇔ x>5

The solutions of this inequality are the numbers greater than 5.

5 is not a solution of the inequality, since 5>5 is false.

In the case of a closed interval and weak inequality, the end-point is a solution of the inequality.

x∈[5;+∞) ⇔ x≥5

The solutions of this inequality are the numbers greater than or equal to 5.

5 is a solution of the inequality, since 5=5 implies 5≥5

In diagrams we mark the end-point of an interval with an empty circle in the case of strict inequality (x>1), and with a filled circle in the case of a weak inequality (x≥1). Illustration

When the boundary is a line, then in the case of a strict inequality (y>x-1) we show it as a broken line (in the program a thin line is used), and in the case of a weak inequality (y≥x-1) we show it as an unbroken (thick) line. Illustration

The notation

y+1∈(5x+1;2y-x]

is equivalent to

y+1>5x+1  AND  y+1≤2y-x


 xy+1∈(5x+1;2y-x]

>,≥,<,≤,≠,ℜ,∈,⇔,⇒,⇐,∨,∧,⊂,∩,∪,∞,∅,∼


•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots

•Simultaneous inequalities

•Inequalities linked with OR


If we add (subtract) the same term to (from) both sides of

the inequality, then the inequality remains equivalent.


If we multiply (divide) both sides of an inequality by a number

greater than zero, then the inequality remains equivalent.


If both sides of an inequality are multiplied (divided)

by a number less than zero and the inequality sign is

reversed then the inequality remains equivalent.


Basic concepts connected with inequalities.
Inequalities -- basic concepts

See also

An inequality consists of two algebraic expressions, at least one of which contains an unknown, joined by one of the signs >,≥,<,≤,≠. Sometimes we write an inequality using several symbols >,≥ or several symbols <,≤ (multiple inequalities) or using the symbol ∈ (belongs to) and intervals.

A solution to an inequality with n unknowns consists of n numbers such that, when they are substituted for the unknowns, a true statement is obtained.

To solve an inequality means to find all the solutions of the inequality.

If an inequality has no solutions then we call it a contradiction.

If an inequality is satisfied by any values of the variables appearing in it then it is called an identity.

Inequalities are called equivalent if they have the same domain and the same set of solutions.

A weak inequality (≥,≤) can be broken down into an equation and a strict inequality.

Theorems on equivalence of inequalities.

•If we reduce similar terms and perform operations appearing on one or both sides of the inequality, then the old and new inequalities are equivalent.

•If we add (subtract) the same term to (from) both sides of an inequality then the old and new inequalities are equivalent.   Example

•If we multiply (divide) both sides of an inequality by a number greater than zero, then the old and new inequalities are equivalent.   Example

•If we multiply (divide) both sides of an inequality by a number less than zero and we also reverse the sign of the inequality, then the old and new inequalities are equivalent.   Example

The signs > and < are the reverse of each other, as are the signs ≥ and ≤. The reverse of the sign ≠ is the same sign.

Example


	An inequality to solve.
	We begin by eliminating the brackets and combining similar terms.
	We now group the terms involving an unknown on the left-hand side, and the numbers on the right. To do this we add, in turn,  -1 and -8x to both sides of the inequality.
	To complete the grouping we combine similar terms.
	We complete the solution by dividing both sides by the coefficient of the unknown (-6), but as this is a negative number we have to reverse the inequality sign.


•Inequalities - basic concepts

•Intervals and inequality signs

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots

•Linear functions


Solution of linear inequalities.
Linear inequalities

See also

A linear inequality is an inequality which can be converted to an equivalent inequality which has a linear function on one side and zero on the other.

	Example inequality to solve.
	We begin by eliminating the brackets and combining similar terms.
	We now group the terms containing an unknown on the left-hand side, and numbers on the right-hand side. To do this we move the appropriate terms to the other side of the inequality, not forgetting to change their signs. We then combine similar terms.
	We divide both sides by the coefficient of the unknown. Since this number is negative, we must reverse the sign of the inequality. We cancel the fraction and write it as a mixed number.
To show the solution on the number line we click the button

Since the inequality is weak, we mark the fact that the end-point is included in the set of solutions with a large spot.


•Inequalities - basic concepts

•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots

•Quadratic functions -definitions

•Graphs of quadratic functions


Solution of quadratic inequalities.
Quadratic inequalities

See also

A quadratic inequality is an inequality which can be converted to an equivalent inequality having a quadratic function on one side and zero on the other.

To solve a quadratic inequality we find the zero points of a quadratic inequality and draw its graph. From the graph we can see for which values the inequality is satisfied.

To solve the inequality

we find the zero points of the quadratic function

and draw its graph. In the program we use Rough Notes to determine the sign of the function.

NB The program does not check Rough Notes.

We enter the result


•Inequalities - basic concepts

•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Quotient inequalities

•Inequalities involving roots


Solution of inequalities of the type x^n<b.
Inequalities of the type

See also

The solution of inequalities of the type

	, where n∈N and n>0,
depends whether n is odd or even.

•For odd n we take the n-th root of both sides.

•For even n

inequality	solution
		•for b≤0    no solutions   exist	•for b>0
		•for b<0    no solutions   exist	•for b≥0
		•for b<0   all x	•for b=0    x≠0	•for b>0
		•for b≤0   all x	•for b>0

Example


We find the solutions of the inequality

depending on the parameters a and b.

We must consider three cases

1) a=0. In this case the inequality is is satisfied by any value of x if only b>0.

2) a>0. Dividing both sides by a we have

This inequality has solutions only if its right-hand side is positive.

Hence, for b>0 we have

3) a<0. Dividing both sides by a we obtain

Depending on the sign of the number on the right-hand side we have:

•for b<0 (right-hand side is positive)	•for b=0 (right-hand side is zero)	•for b>0 (right-hand side is negative)
	x≠0	all x


•Inequalities - basic concepts •Intervals and inequality signs •Linear inequalities •Product inequalities of the type ab>0 •Linear inequalities with two unknowns •Quadratic inequalities •Inequalities of the form •Quotient inequalities •Inequalities involving roots		•Equations - basic concepts •Linear equations •Equations in many variables •Product equations ab=0 •Quadratic equations •Biquadratic equations •Symmetric equations •Quotient equations •Equations of the form •Use of the factor theorem in solving equations •Equations involving roots


Solution of inequalities in one variable.
Inequalities in one variable -- advanced level

See also

•To solve the inequality

f(x)≠0,  where f is a function,

we determine the domain of the function f, we solve the equation

f(x)=0

and negate its solutions.

Easier and harder examples

•To solve the inequality

f(x)<0,  where f is a function,

we determine the domain of the function f, find its zero points, sketch the graph and read off from the graph the intervals on which the inequality holds.

Obviously we proceed analogously when we have other signs: >,≥,≤.

Examples


•To solve the inequality		•To solve the inequality
1.			2.
we solve the equation		we solve the equation
•			•
and negate its solution		and negate its solution
•			•
		NB The negation of a disjunction of logical statements is the conjunction of their negations.


•To solve the inequality		•To solve the inequality
1.			2.
we find the domain		we find the domain
•			•
solve the equation		solve the equation
•			•
negate its solution		negate its solution
•			•
                    and combine this with the domain definition

•			•


•We shall solve the inequality

The expression on the left-hand side of the inequality is a biquadratic function of the variable x. Finding its zero points gives:

We sketch the graph. In the program we use Rough Notes for this (the program does not check Rough Notes). The solution of our inequality is therefore

•We now solve the inequality

We write the left-hand side as a product.

If some factor is always positive or always negative then we can divide both sides by this factor.

Notice that [image]. After dividing on both sides we have:

After dividing both sides by (-3) we obtain:

We sketch the graph		Hence
•We shall now solve the inequality
We do this by sketching the changes of sign undergone by the expressions in brackets. In blue we sketch the sign changes for the function y=1-x (for x<1 it is positive, for x=0 it is equal to zero, and for x>1 it is negative). In pink we mark the sign changes for the function  y=x-3 (repeated three times, as it is cubed).
Hence


•Inequalities - basic concepts

•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Inequalities involving roots


Solution of quotient inequalities.
Quotient inequalities

See also

Inequalities of this type can be solved using the following theorems:

To solve the inequality

we change the quotient into a product, using the above theorem

We solve the simultaneous inequalities obtained

	Hence x∈(-5;2]


•Inequalities - basic concepts

•Intervals and inequality signs

•Linear inequalities

•Product inequalities of the type ab>0

•Linear inequalities with two unknowns

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities


Solution of inequalities involving roots.
Inequalities involving roots

See also

••The solution of inequalities of the type

	, where n∈N and n>0,
depends whether n is odd or even.

•For odd n we raise both sides to the power n.

•For even n

inequality	solutions
		•for b≤0    no solutions   exist	•for b>0
		•for b<0    no solutions   exist	•for b≥0
		•for b<0   x≥0	•for b=0   x>0	•for b>0
		•for b≤0   x≥0	•for b>0
	The condition x≥0 is the condition defining the domain of the root.
Example

••Sometimes inequalities involving roots can be solved by substitution.

Example


To solve the inequality

we make the substitution

to convert it into a quadratic inequality

After restoring the original variable we obtain

This gives


•Substitution method

•Elimination method

•Graphical method


Basic concepts connected with simultaneous equations.
Simultaneous equations -- basic concepts

See also

If we have n equations and we want to find solutions satisfying all of them, then we say that we are solving n simultaneous equations.

A solution of a set of simultaneous equations in n unknowns consists of n numbers which, when substituted for the unknowns, satisfy all of the given equations.

For simultaneous equations with two unknowns a solution will be a pair of numbers, when there are three unknowns the solution will be a triple of numbers, etc.

To solve a set of simultaneous equations means to find all the solutions for the given set of equations.

A set of simultaneous equations may have no solutions, one solution, two, ... , or infinitely many solutions.

The domain of a set of equations is the intersection of the domains of the individual equations.


The following simultaneous equations have no solution

If we substitute the value of y from the first equation into the second, we get

We find y from both equations and draw the lines on a coordinate diagram, we can see that the two lines are parallel, which means there are no solutions satisfying both equations simultaneously.


To solve the simultaneous equations

we start by solving the first of the equations.

This means that there are two solutions


We solve the simultaneous equations

1.		2.		3.

As can be seen, there is only one solution.

If we calculate y from both equations and draw the lines on a coordinate diagram, we see that they meet at one point. The coordinates of this point are the solution of the set of equations.


To solve the simultaneous equations

we notice that, when the value of x from the second equation is substituted into the first equation, we get the identity 0=0.

Therefore all pairs of numbers which satisfy the second equation are solutions for the set of simultaneous equations. There are infinitely many of them.

If we find y from both equations and draw the lines on a coordinate diagram, we see that both equations give the same line.


To find a given variable in terms of the others, we treat all other variables like numbers and proceed as we do when we solve equations in one unknown.

To find an expression for y
we group all terms containing y on the left-hand side of the equation and the remaining terms on the right-hand side. When moving terms from one side of the equation to the other we must remember to change the sign.
As we have found -y we must multiply both sides of the equation by (-1)


•Simultaneous equations -- basic concepts

•Elimination method

•Graphical method


Substitution method - example

To solve the simultaneous equations

evaluate y from the first equation.
We now substitute this value for y into the second equation
and evaluate x.
We substitute the value found for x back into the first equation, and get


The substitution method for solution of simultaneous equations
Substitution method

See also

The substitution method for solving simultaneous equations involves finding an expression for one of the variables from any one of the equations, and then substituting it into the other equations.

Example

•If while solving simultaneous equations we obtain a contradiction (eg.. 2=3) then we know that there can be no solutions. In the program we click the button

•If all variables in all the equations can be eliminated, and the remaining equations are all true statements (eg. 0=0, 2=2) then we know that our equations are satisfied by any values of the variables appearing in them (identity). In the program we click the button

NB  The simultaneous equations

do not constitute an identity.


•Simultaneous equations -- basic concepts

•Substitution method

•Graphical method


Elimination method - example

We solve the simultaneous equations

	We make the coefficients of  x correspond to each other		We make the coefficients of y correspond to each other
	Next we add both sides of the equations. As the other simultaneous equation we have chosen the previous equation II.		Now we add both sides of the equations. As the other simultaneous equation we have chosen the previous equation I.
	From the first equation we find y		From the first equation we find x
	and substitute this value into the second equation		and substitute this value into the second equation
	After finding x from the second equation we obtain the final solution		After finding y from the second equation we obtain the final solution


The elimination method for solving simultaneous equations
Elimination method

See also

The elimination method allows us to bring the coefficients of one particular variable in two simultaneous equations to the same absolute value, though they have opposite signs. Next we replace one of the equations by the result of adding the equations.

Example

•If while solving simultaneous equations we obtain a contradiction (eg. 2=3) then we know that there are no solutions. In the program we click the button

• If all variables in all the equations can be eliminated, and the remaining equations are all true statements (eg. 0=0, 2=2) then we know that our equations are satisfied by any values of the variables appearing in them (identity). In the program we click the button

NB  The simultaneous equations

do not constitute an identity.


The coefficients of x are 2 and 4.

We want them to have the same absolute value,

so we find their least common multiple

(as for finding the lowest common denominator of fractions)

                   LCM(2, 4)=4.

We therefore multiply the first equation by 2, since (2∗2=4).

In order to give the coefficients opposite signs we have to multiply one of the equations by (-1)

To avoid performing two separate multiplications, we can multiply the first equation by (-2).


The coefficients of y are 3 and 2.

We want them to have the same absolute value,

so we find their lowest common multiple

(as for finding the lowest common denominator of fractions)

                   LCM(3, 2)=6.

We therefore multiply the first equation by 2 (since 3∗2=6),

and the second equation by 3 (since 2∗3=6).

Note that the signs of the coefficients of

y are opposite.


Adding two equations involves adding the two left-hand sides and adding the two right-hand sides.

This reduces to


Adding two equations involves adding the two left-hand sides and adding the two right-hand sides.

This reduces to


Adding two equations involves adding the left-hand sides and adding the right-hand sides of the equations.


After adding the equations, we can chose either of the equations we added as the other simultaneous equation. It can be used either in its original form or in the form used for the addition.


To find a given variable in terms of the others, we treat the other variables as numbers and proceed as if we were solving an equation in one unknown.

We find y from both equations.

Remember to change the sign when moving terms from one side of the equation to the other.


•Simultaneous equations -- basic concepts

•Substitution method

•Elimination method

•The graph of a linear function


Graphical solution of simultaneous equations
Graphical method

See also

We shall describe the graphical method for solving two simultaneous linear equations in two unknowns.

To solve graphically the simultaneous equations

we evaluate y from both equations.

We now draw the two lines.

	The point of intersection of the two lines is the solution of the equations. In this case the solution is the point  (1; 2) where the two graphs meet. The solution can also be written as follows                    x is the first coefficient of the point, y is the second coefficient of the point.

•If the graphs do not all intersect at one point then the simultaneous equations have no solution.  Illustration

•If the graphs are identical then the simultaneous equations have infinitely many solutions.  Illustration


If the graphs do not all intersect at one point then the simultaneous equations have no solutions.


If all the graphs are identical then the simultaneous equations have infinitely many solutions.


Example of a set of simultaneous inequalities having no solutions.


Example of a set of simultaneous inequalities which holds everywhere.

	The inequalities hold for any values of x,y.


•Simultaneous inequalities in one unknown

•Simultaneous linear inequalities with two unknowns

•Estimation -- advanced level


Basic definitions concerning simultaneous inequalities.
Simultaneous inequalities -- basic concepts

See also

A set of n simultaneous inequalities consists of n inequalities joined by AND. We usually brace the inequalities together  and omit the connective(s) AND.

The domain of a set of simultaneous inequalities is the intersection of the domains of the individual inequalities.

A solution of a set of simultaneous inequalities in p unknowns consists of p numbers satisfying all the inequalities.

To solve a set of simultaneous inequalities means to find all of its solutions.

If a set of simultaneous inequalities has no solutions, then we say it is contradictory.

If a set of inequalities holds for all values of the variables appearing in it  then we call it an identity.


In the window	In the window
•We solve the simultaneous inequalities	•we use a diagram to help us. (We write the inequalities using connectives rather than bracing them together.) •hence
•	•since there are no numbers satisfying both conditions simultaneously.
•we click the button


In the window	In the window
•We solve the simultaneous inequalities •The result can be written either in a brace or as an interval.	•to check whether the conditions can be written more simply we use a diagram. (We write the inequalities using connectives, rather than bracing them together.) •It can be seen that neither condition is redundant.


In the window	In the window
•We solve the simultaneous inequalities	•to check whether the conditions can be written more simply we use a diagram. (We write the inequalities using connectives rather than bracing them together.) •It can be seen that the condition x≤1 is not needed.
•The solution is therefore x<0


•Simultaneous inequalities - basic concepts

•Simultaneous linear inequalities in two unknowns

•Estimation -- advanced level


Solution of simultaneous inequalities in one variable.
Simultaneous inequalities in one unknown

See also

To solve a set of simultaneous inequalities in one unknown, we solve each inequality separately and then mark the results on the number line. We can then check for which values of the variables all the inequalities are satisfied simultaneously.

The solution can often been written in several equivalent ways:

Example  of a contradictory set of inequalities

Example of a set of simultaneous inequalities with a redundant condition

Example 3


Example of a contradictory set of inequalities

If we draw the regions represented by these inequalities on a diagram, we can see that the intersection is the empty set.

y<x+1 AND x<2 AND x>5	There is no region which is shaded in all the colors.


We solve the simultaneous inequalities:

	After finding y from each inequality we mark the regions on a coordinate diagram y<2x  AND  y<x  AND  y>-x
	The region which contains shading in all the colors is the set of points which satisfy the simultaneous inequalities.

It can be seen that the inequality y<2x is not needed.

Therefore the solution for the simultaneous inequalities is

	⇔  y<x  AND  y>-x


We find y from each inequality. If the variable y does not appear in an inequality then we evaluate the variable x in that inequality.


•Inequalities - basic concepts

•Linear inequalities

•Product inequalities of the type ab>0

•Inequalities in one variable - advanced level

•Quadratic inequalities

•Inequalities of the form

•Quotient inequalities

•Inequalities involving roots


Solution of linear inequalities in two variables.
Linear inequalities in two unknowns

See also

In a coordinate diagram, the solution of a linear inequality in two unknowns can be represented by a half-plane. The boundary line of this region is contained in the region (is a solution of the inequality) if the inequality is weak (≥,≤); this is shown with a thick line.

If we have an inequality in x and y we usually find y. When finding y in terms of x, we treat x as a number and proceed as for solving an inequality in one unknown.

The solution of the inequality y-2x-1>0 consists of all pairs of numbers (x;y) such that y>2x+1. To draw this on the coordinate diagram we draw the line y=2x+1, and shade the half-plane above this line. Illustration

If the inequality had been weak (≥) then points on the line y=2x+1 would also have belonged to the set of solutions of the inequality. Illustration

In general we have:

a>0	a<0	a=0
In the program, the window for marking regions is opened by clicking on the button


The regions where the respective inequalities are satisfied are:

a>0	a<0
For one variable we have:


•Simultaneous inequalities - basic concepts

•Simultaneous inequalities with one unknown

•Estimation -- advanced level

•Linear inequalities with two unknowns


We shall solve the simultaneous inequalities:

	We evaluate the variable y from the inequalities in which it appears, and x in the others.
We now mark the regions on a coordinate diagram.

In the program, the window for marking regions is opened by clicking on the button [image] and the conditions we wish to draw should be joined with the connective AND, not braced together.

y≤4  AND  x≥1  AND  y>x-1

	The region which is shaded in all the colors is the set of points which satisfy the simultaneous inequalities.
	It can be seen that none of the conditions is redundant. This is therefore the final solution.


Solution of simultaneous linear inequalities in two unknowns.
Simultaneous linear inequalities in two unknowns

See also

To solve a set of simultaneous linear inequalities we decide which variable we are going to evaluate, and evaluate it in each of the inequalities in which it appears. Next we mark the regions on a coordinate diagram, and check for which values all the inequalities are satisfied simultaneously.

Example 1.

Example of a contradictory set of inequalities (having no solutions)

Example  of a set of simultaneous inequalities with a redundant condition.


We wish to solve the simultaneous inequalities

Note that

Hence the inequality

is redundant, and the solution reduces to


We wish to solve the simultaneous inequalities

Note that

the inequality holds for any value x

Which means that the inequality

is redundant and our solution reduces to the inequality


We wish to solve the simultaneous inequalities

Notice that

Hence there are no solutions.


•Simultaneous inequalities - basic concepts

•Simultaneous inequalities with one unknown

•Simultaneous linear inequalities with two unknowns


Methods of estimating inequalities.
Estimation -- advanced level

See also

We let f and g represent functions of the variable x.

When estimating inequalities we use the following theorems:

•For the simultaneous inequalities:  •If for any x in the domain we have f(x)<g(x) then the condition y<g(x) is redundant and can be deleted. •If there is no x in the domain for which f(x)<g(x) then the condition y<f(x) is redundant and can be deleted. Example		•For the simultaneous inequalities:  •If for any x in the domain we have f(x)>g(x) then the condition y>g(x) is redundant and can be deleted. •If there is no x in the domain for which f(x)>g(x) then the condition y>f(x) is redundant and can be deleted. Example
•The set of inequalities		can be written f(x)>y>g(x).

The simultaneous inequalities therefore have solutions only when x satisfies f(x)>g(x).

Example


The gray stripe below the shaded regions shows the set created by combining the red and blue regions. Hence

x∈(2;10]


We can see that the combination of the red and blue regions (represented by the gray stripe) can be written:

x∈ℜ


We can see that the combination of the red and blue regions (represented by the gray stripe) can be written:

x<5


The conditions in brackets are marked in red, and the remaining condition in blue. Note that the red and blue conditions are joined with AND, so we have to look for the region where both red and blue conditions are satisfied. On the diagram this is represented by the gray stripe. Hence

x∈(4;8)


The conditions from the first bracket are marked in red, and those from the second bracket in blue. Note that the red and blue conditions are joined with AND, so we have to look for the region in which both red and blue conditions are satisfied. This is represented on the diagram by the gray stripe. Hence

x<-1 ∨ x∈(0;1) ∨ x>2


We mark the conditions on a coordinate diagram

To make it easier, we first mark the region satisfying the conditions in brackets (red and light blue)		Outside the bracket we have the connective AND, so we must choose the intersection of the region represented by the bracketed conditions with the region represented by the remaining condition (dark blue).
		It can be seen that to define the marked region it is enough to state the following conditions: y<2x AND x≥0


•Simultaneous inequalities

•Union (∪) and intersection of sets (∩)

•Disjunction (∨,OR) and conjunction (∧,AND) of statements

•Cases


Solution of inequalities joined by the connective OR
Inequalities linked with OR.

See also

•If we have inequalities in one variable joined by the connective ∨ (OR) then we mark the solutions of all the inequalities on the number line and check whether the intervals obtained can be written using fewer conditions.

•x∈(2;4) OR x∈[3;10] ⇔ x∈(2;10]

Illustration

•x<9 OR x>5 ⇔ x∈ℜ

Illustration

•x<5 OR x<4 ⇔ x<5

Illustration

•(x<-1 ∨ x>4) ∧ x∈[2;8) ⇔ x∈(4;8)

Illustration

•(x<1 ∨ x>2) ∧ (x<-1 ∨ x>0) ⇔ x<-1 ∨ x∈(0;1) ∨ x>2

Illustration

•We proceed similarly when we have inequalities in two variables, except that their solutions are shown on a coordinate diagram.

•y<2x+1 OR y=2x+1 ⇔ y≤2x+1

•(y<2x ∨ y<x) ∧ x≥0 ⇔ y<2x ∧ x≥0

Illustration

Remember that

x∈A OR x∈∅ ⇔ x∈A x∈A ∨ x∈∅ ⇔ x∈A x∈A OR x∈ℜ ⇔ x∈ℜ x∈A ∨ x∈ℜ ⇔ x∈ℜ		x∈A AND x∈∅ ⇔ x∈∅ x∈A ∧ x∈∅ ⇔ x∈∅ x∈A AND x∈ℜ ⇔ x∈A  x∈A ∧ x∈ℜ ⇔ x∈A
x∈∅ denotes a contradiction.

In the program, the window for marking regions is opened with the button

Logical connectives can be written either in words or using symbols, depending on the option selected.


Mathematical symbols

∪ -- union of sets ∩ -- intersection of sets	∨, OR -- disjunction of statements  ∧ ,AND -- conjunction of statements


•Simultaneous inequalities

•Inequalities linked with OR

•Disjunction (∨,OR) and conjunction (∧,AND) of statements

•Cases


Definitions and laws concerning unions and intersections of sets.
Union and intersection of sets

See also    Mathematical symbols

The union of two sets (∪) is the set of all the elements contained in either of them.

x∈A∪B ⇔ x∈A ∨ x∈B

The intersection of two sets (∩) is the set of those elements which are common to both sets.

x∈A∩B ⇔ x∈A ∧ x∈B

Remember that

A∪B=B∪A A∪∅=A A⊂B ⇒ A∪B=B		A∩B=B∩A A∩∅=∅ A⊂B ⇒ A∩B=A

(A∪B)∩C=(A∩C)∪(B∩C)

(A∩B)∪C=(A∪C)∩(B∪C)

Order of operations: brackets, intersections (∩), unions (∪).

A∪B∩C = A∪(B∩C)

Examples

(3;5) ∪ [4;7] = (3;7],  since

   x∈(3;5) ∪ [4;7] ⇔ x∈(3;5) ∨ x∈[4;7] ⇔ x∈(3;7]

(3;5) ∩ [4;7] = [4;5),  since

   x∈(3;5) ∩ [4;7] ⇔ x∈(3;5) ∧ x∈[4;7] ⇔ x∈[4;5)


•Simultaneous inequalities

•Inequalities linked with OR

•Union and intersection of sets

•Cases


The set of statements written using the connective OR

can be written as the union of three cases.

Case 1.	Case 2.	Case 3.

In the program we set up a new case by using the keys SHIFT+Y.


The notation is equivalent to		The statements are equivalent to


Given the statements

We can write them as follows

Now we can break them down into two cases (SHIFT +Y)

Case 1.	Case 2.
We must break down each case further.

Case 1.1.	Case 1.2.	Case 2.1.	Case 2.2.


Examples

•∼(x≥1) ⇔ x<1

•∼(x<6) ⇔ x≥6

•∼(x≠3) ⇔ x=3

•∼x∈[2;5) ⇔ ∼(x≥2∧x<5) ⇔ ∼(x≥2)∨∼(x<5) ⇔ x<2∨x≥5

•∼x∈(1;6)∪[8;+∞) ⇔ ∼(x∈(1;6)∨x∈[8;+∞)) ⇔ ∼(x∈(1;6))∧∼(x∈[8;+∞)) ⇔

⇔ (x≤1∨x≥6)∧∼(x≥8) ⇔ (x≤1∨x≥6)∧x<8 ⇔ (x≤1∧x<8)∨(x≥6∧x<8) ⇔

⇔ x≤1∨x∈[6;8) ⇔ x∈(−∞;1]∪[6;8)


Laws concerning logical statements (AND, OR, FALSE).
Disjunction and conjunction of logical statements

See also    Mathematical symbols

•The disjunction of two statements (∨, OR) is true when at least one of the two statements is true.

•The conjunction of two statements (∧, AND) is true if and only if both statements are true.

•The negation (∼) of a true statement is false.

•The negation (∼) of a false statement is true.

If x is a number, then

∼x∈∅ ⇔ x∈ℜ

∼x∈ℜ ⇔ x∈∅

Order of operations: brackets, conjunctions (∧, AND), conjunctions (∨, OR).

p∨q∧r ⇔ p∨(q∧r)

Distributive laws:

(p∨q)∧r ⇔ (p∧r)∨(q∧r)

(p∧q)∨r ⇔ (p∨r)∧(q∨r)

de Morgan's laws:

∼(p∨q) ⇔ ∼p∧∼q

∼(p∧q) ⇔ ∼p∨∼q

Pairs of complementary signs are: < and ≥,  > and ≤,  ≠ and =

Examples

If we have conditions joined by the connective ∨ (OR) then we can write them as separate cases. Example

If we have conditions joined by the connective ∧ (AND) then we can write them using a brace or add them to the existing conditions as a new line. Example

If we have conditions containing both ∨ (OR) and ∧ (AND) then, when breaking them down into cases, we must remember about the order of operations, and operations which change the notation must sometimes be repeated several times. Example


•Inequalities - basic concepts

•Equations - basic concepts

•Simultaneous equations --basic concepts

•Simultaneous inequalities - basic concepts


Contradictions and identities.
Contradictions and identities

See also

An identity is a statement which is always true. 0=0 2=2 x=x 3>-2 y≥y	A contradiction is a statement which is always false. 0>0 2≠2 x>x 3≤-2 y<y

If an equation has no solutions then it is called contradictory.

If an equation holds for any values of the variables appearing in it then it is called an identity.

If an inequality has no solutions then it is called contradictory.

If an inequality is holds for any values of the variables appearing in it then it is called an identity.

Example 1.		Example 2.
To solve the simultaneous equations we note that the second equation is an identity. We can therefore delete it.		To solve the simultaneous equations we notice that the first equation is a contradiction. There are therefore no solutions to the simultaneous equations.


•Equations - basic concepts

•Linear equations

•Quadratic equations

•Biquadratic equations

•Symmetric equations

•Quotient equations

•Equations of the form

•Use of the factor theorem in solving equations

•Equations involving roots


Equations in many variables.
Equations in many variables

See also

If an equation contains more than one variable, then we solve it as f the variable we want to find were the unknown, and the other variables were ordinary numbers.

•	4x+3y-6=2x+y	We wish to find y. We group the terms containing y together on the left-hand side of the equation, and the remaining terms on the right-hand side.
	3y-y=2x-4x+6	We combine similar terms on each side.
	2y=-2x+6	We divide both sides by 2
	y=-x+3
•	yx=1	If we wish to divide both sides by x in order to find y, then we must consider the cases x=0 and x≠0.
		1.	x=0	2.	x≠0
			y∗0=1 0=1 contradiction


•Biquadratic equations

•Symmetric equations

•Equations involving roots


Substitution

See also

Substitution involves replacing some algebraic expression by an auxiliary variable.

The auxiliary variable must be given a name which does not appear anywhere else in the solution, in order to avoid confusion between variables.

In the program, the definition of an auxiliary variable is always placed on a new line by itself.

To distinguish the definition from other equations we use the substitution symbol ←( CTRL+ =); in other places we use the normal equality sign =.

Example

We wish to solve the equation

We restore the original variable

NB The program does not allow the definition of an auxiliary variable

to contain the variable itself (eg. z←5z+1)

nor can an implicit definition be used (eg. 2z←x+1).


•Inequalities linked with OR

•Simultaneous equations

•Simultaneous inequalities

•Disjunctions (∨,OR) and conjunctions (∧,AND) of statements


The disjunction of the conditions

x<4 OR x≥8

can be written in the firm of cases:

Case 1 x<4	Case 2 x≥8
and further:

Case 1 x<4	Case 2 x>8	Case 3 x=8


The statements

can be written as a combination of two cases

Case 1	Case 2
Then each case can be solved separately

Case 1	Case 2 no change
After this (if necessary) we can rewrite them in the same window. In the program we use SHIFT+X.


Breaking down problems into cases.
Cases

See also

If we have inequalities of equations joined by ∨ (OR) then we can write them in the form of the appropriate number of cases.

In the program we open a new case using the keys SHIFT+Y.

Note Conditions joined by ∧ (AND) we brace together. This is a set of simultaneous equations or inequalities.

Example 1.

Example 2.


•Linear equations

•Linear inequalities

•Equations -- basic concepts

•Inequalities -- basic concepts


Moving terms to the other side of an equation or inequality.
Moving terms to the other side of an equation or inequality

See also

According to the theorems on equivalent transformations of equations and inequalities, if we add to (subtract from) both sides of an equation or inequality the same monomial then the solution does not change.

y+2x=3x | -2x	To group the terms containing x on the right-hand side of the equation, we add to both sides -2x
y+2x-2x=3x-2x	We reduce similar terms on the left-hand side of the equation.
y=3x-2x	Notice that the equation looks the same as it would if we had moved the term +2x from one side of the equation to the other, and reversed its sign to -2x
y=x


When we perform operations on both sides in the program,

we cannot change the order of the equations (inequalities).

We mark our intention of operating on both sides	The operations have been carried out correctly	The operations have been carried out correctly. We have added the domain, but our equation is still the first to appear.
	If we write it like this, then the program will consider the operation to have been carried out incorrectly. Our equation was the first, so the program will check the equation which now appears first to see if the operation was done correctly.


When operating on both sides, the program does not allow us to swap the sides of the equation (inequality).

If we indicate our intention to operate on both sides

2<-x  |∗(-1)

then the only correct way of doing this in the program is

-2>x,

because the computer checks the two left-hand sides to see if the operation has been carried out correctly, then it checks the right-hand sides and possibly a change of sign.

Obviously x<-2 is also correct and we can use the two forms interchangeably except in the situation described above.


If we indicate our intention of operating on both sides

2-x=-3x  |+3x

but do not do the operation, leaving the equation as it is

2-x=-3x,

then the program will report an error, since it will have compared the left-hand sides and right-hand sides to see if the operation has been carried out correctly.


We wish to solve the inequality

We shall multiply both sides by x. We break the problem down into three cases:

(it is not the recommended method of solution)

1.	contradiction
2.
3.
Hence the solution of the example is the set x<0 or x≥3

The recommended method of solution is as follows:

	We use the theorems on the signs of products and quotients.


We wish to solve the equation

We will multiply both sides by x. We break down the problem into two cases:

(it is not the recommended method of solution)

1.
2.
Hence x=3.

The recommended method of solution is as follows:

	We use the theorems on the signs of quotients and products.


Operations on both sides.
Operating on both sides

Cases

Theorems concerning operations on both sides can be found in the following chapters:

•Equations -- basic concepts

•Inequalities -- basic concepts

In the program it is very important, when operating on both sides:

•not to change the order of the equations (inequalities) -- the part containing the domain is treated separately from the part containing the simultaneous equations or inequalities, example;

•not to swap the sides of the equation, example;

•to actually carry out the operation indicated, example.

If when we carry out an operation we restrict the domain, then we must consider the remaining part of the domain as a separate case, for example if we want to add [image] and the domain is not yet restricted to a≥0 then we have to consider two cases (a≥0 and a<0).

If we want to multiply both sides by a variable or an expression containing one, then we must remember that:

→ we can multiply both sides of an equation only by a non-zero number, so the problem should be broken down into cases in order to check what happens when the expression we are multiplying by is equal to zero,

→ when we multiply both sides of an inequality, we have to know the sign of the number we are multiplying by, so we should consider separately the cases where it is less than zero and where it is greater than zero.

The recommended method of solution in such cases is to take all terms onto one side, put them over a common denominator and change them to quotient form (Quotient equations, Quotient inequalities).

Example of an equation  and an  inequality.


Error in the solution.
Error in the solution

Expansion and restriction of the domain    Increasing and reducing the number of solutions

•If we have selected STRICT DOMAIN CHECKING then the message "Error in the solution" refers only to transformations and solutions outside the domain definition, as the correctness of transformations inside the domain definition is checked earlier and any errors reported separately.

•If STRICT DOMAIN CHECKING is not selected, then the error should be looked for both inside and outside the domain definition, as the program does not check the domain separately.


Error in the domain definition

Domain of a rational expression    Domain

The message "Error in the domain definition" means that:

•you have made an error in transforming expressions, solving equations or inequalities in the Domain section, or

•you have performed a transformation of the expression outside the domain definition which affects the domain.

Example 1. While solving the equation

we canceled the fraction thus:

It is clear that the domains of these expressions are not the same.

To correct the error we must add the condition defining the domain.

	In a case like this it is enough just to write in x≠1, as the computer will add the condition  x≠-1 itself (the denominator must always be non-zero). If however we add the condition  x≠-1 to the domain then it will not be an error.
	After further transformations, the domain definition must contain both conditions  x≠1 AND x≠-1

Example 2. If we transform the expression

and make a mistake in the denominator of the second expression

then the domains are clearly not the same. The domain of the first expression is defined by the inequalities

a≠-1 AND a≠3

while the second domain is given by

a≠-1 AND a≠0.


•Domain of a rational expression

•Domain

•Cases


Expansion and restriction of the domain.
Expansion and restriction of the domain

See also

Expansion of the domain means adding elements to it that were not contained in it before; restriction of the domain means removing certain of its elements.

Example 1. (expansion)

Example 2. (restriction)

Example 3.


Expansion of the domain -- example.

Domain of a rational expression

We have STRICT DOMAIN CHECKING selected.

If we write the equation

(A)
in the form

(B)
then we have expanded the domain.

The expression A only makes sense when

x≠0,

while the expression B makes sense for all

x∈ℜ,

also for x=0.

Hence the domain of the expression A is contained in the domain of the expression B.

{x∈ℜ; x≠0}⊂ℜ

The correct entry of B is


Restriction of the domain -- example.

Domain of a rational expression     Cases

If we have STRICT DOMAIN CHECKING selected and we expand the fraction in the following way

then we have restricted the domain.

The domain of the expression A is given by the condition

x≠1,

whereas the domain of the expression B is given by

x≠1 AND x≠-1.

Hence the domain of the expression B is contained in the domain of the expression A.

 {x∈ℜ; x≠1 AND x≠-1}⊂{x∈ℜ; x≠1}

In such a case, in order to make the domain correct we must consider two cases:


Domain of B case = Domain of B1 case ∪ Domain of B2 case =

	{x∈ℜ; x≠1 and x≠-1}∪{x∈ℜ;x=-1} =  {x∈ℜ; (x≠1 and x≠-1) or x=-1} =  {x∈ℜ; x≠1 and (x≠-1 or x=-1)} =  {x∈ℜ; x≠1 and x∈ℜ} =  {x∈ℜ; x≠1} =
Domain of A case


Notice that we can also alter the domain when we make a mistake with our transformations. If we transform the expression

and make a mistake in the denominator of the second expression

then clearly the domains are not the same. The domain of the first expression is defined by the conditions a≠-1 AND a≠3 while the second domain is given by a≠-1 AND a≠0.


Increasing and reducing the number of solutions.

Domain of a rational expression    Cases

The message "You have increased the number of solutions'' or "you have extended the solution set" means that the previous set of solutions was a subset of the new set of solutions. (Previous solution set⊂New solution set)

The message "You have reduced the number of solutions'' means that the new solution set is a subset of the previous solution set. (New solution set⊂Previous solution set)

NB We take into account only those solutions which lie in the domain.

The above messages may appear when:

•in Transformations, we have extended (restricted) the domain

•in Equations, we have extended (restricted) the domain or the solution set of the equation or of the simultaneous equations or inequalities.

Example 1.

Example 2.

Example 3.

Example 4.


If you introduced an auxiliary variable using the symbol = instead of ←  then the program will treat this as a reduction in the number of solutions. Notice that since this variable does not appear in the previous stages, it can take any value.

After an auxiliary variable is defined, you can treat it like any other variable.


Reducing the number of solutions -- example.

Cases

If we have not selected STRICT DOMAIN CHECKING and we expand the fraction in the following way

then we are reducing the number of solutions.

The domain of the expression A was given by the condition

x≠1

while the domain of the expression B is the set given by

x≠1 AND x≠-1.

So we have lost the solution for x=-1.

To avoid the mistake two cases must be considered:


If we solve the inequality x(x-1)>0

and give as the solution x<0 then we

have reduced the number of solutions,

as the correct answer is x<0 OR x>1.

{x∈ℜ; x<0}⊂{x∈ℜ; x<0 OR x>1}


We are solving the equation x(x-6)=0 with the domain condition x>5.

If we give as the answer x=0 OR x=6 then we have clearly increased the number of solutions, as the correct answer, taking the domain into account, is x=6.


•Intervals and inequality signs

•Linear inequalities with two unknowns

•Simultaneous linear inequalities with two unknowns

•Inequalities linked with OR


Wrong region marked.
Wrong region marked

See also

The message "Wrong region marked" appears when:

•we have marked the region on the wrong side of the point (line)

•we have not taken the boundary into account with weak inequalities (≥,≤)

 Illustration 1     Illustration 2     Illustration 3     Illustration 4

•we have taken the boundary into account with strict inequalities (>,<)

•we have included too many conditions in the final answer

•we have not included all conditions in the final answer

NB When we give the final answer we must indicate which of the points marked with an empty circle or which of the lines on the graph belong to our region. We do this by clicking on them; clicking on them again unmarks what was already marked.


Something is missing.
Something is missing

The message "Something is missing" appears when a sign is missing (eg. between two numbers) or an expression is missing (eg. in the numerator or denominator of a fraction, or between opening and closing brackets when there is nothing there). Usually in a place like this the computer puts a question mark -- this should not be deleted, instead the missing expression or sign should be entered.

If while solving an example we reach a position where all equations and inequalities outside the domain definition are identities, then:

•if there are no conditions in the domain we click the identity button,

•if there are conditions in the domain, then we move them to the solution part, and either continue solving or click the answer button as appropriate.


It was much better before.
It was much better before

The message "It was much better before, now it's too complicated, I won't check it!" appears when previously the example could be solved, but now the numbers are too big or the computer has exceeded the maximum number of operations.

We are solving a physics problem.

The computer finds the value of z from the equation.

If the user finds the value of w

then unfortunately the computer will show the above message (the maximum number of operations has been exceeded ninefold). In a case like this, if the problem includes numerical data, then it is good to enter this data in the window and not to select STRICT DOMAIN CHECKING.


The symbols cannot be placed together. This symbol may not appear in this position.
The symbols cannot be placed together

The message "The symbols cannot be placed together" appears when signs which should not appear next to each other nevertheless do so.

Examples:

+-3  should be eg. +4-3 or perhaps +(-3)

(;3)  should be eg. (4;3)

x>(3;6)  should be eg. x>4 OR x∈(3;6) or perhaps x>4 AND x∈(3;6)

The message "This symbol may not appear in this position" means that you must complete or alter the notation on the left-hand or right-hand side of the cursor so that it makes sense (as in the case of "The symbols cannot be placed together"). Usually the problem occurs with notation involving brackets and intervals, logical statements and sets.

Example

x>4(3;6)  should be eg. x>4 OR x∈(3;6)


Too many operations.
Too many transformations

 The program is equipped with a protective function which causes it to stop working when it has too many operations to consider. Calculations are carried out symbolically, so when the number of variables is too large or an example is too difficult, it would take too long to complete them.

If the problem includes amounts which are parameters that the computer does not yet know about, then enter them in the DATA AND AUXILIARY VARIABLES window, each parameter on a separate line.


Too large numbers

The message "Too large numbers" appears when the computer's calculations give rise to a number whose absolute value is greater than

or smaller than


Hints.
Hints

The program gives localized hints, that is to say if we ask for a hint at the sign:

= we find out how to solve an equation;

>,≥,<,≤ we find out how to solve an inequality;

 +,- we find out if we can perform reduction, in which expressions making up the sum there is still something to be done.

Similarly, at

* we receive hints concerning multiplication of expressions, at the

fraction line we find out, for example, whether the fraction can be canceled, and at the

root sign we find out whether it can be calculated or whether a factor can be taken outside the root etc.

Note 1  When the computer gives hints it does not check whether the transformations already performed are correct. The only way to find out if there has been a mistake in the calculations is by pressing ENTER.

Note 2  The program often gives several hints, so it is best to read them all and choose the ones which apply to your strategy for solving the problem.

The message "Ask for a hint elsewhere" appears when the program cannot see anything to do in the position given.


Transformations of expressions
Transformation of expressions

The transformation of an expression involves:

•simplifying the expression

•performing operations on expressions. See:

   ••Monomials and polynomials

   ••Rational expressions - definitions and operations

   ••Absolute value (modulus) of an expression

   ••Powers

   ••Roots

   ••Multiplication

NB If we are in the window TRANSFORMATIONS then it is best to select STRICT DOMAIN CHECKING. This is especially important in problems containing the instruction "Transform the expression and then calculate its value for ..." and "State the domain and transform the expression".

An exception is problems with many variables which the computer decides are too difficult for it. Then, if we have data for the problem, if STRICT DOMAIN CHECKING is switched off we will at least know if the result is correct after substitution of data.


Equations and inequalities.
Equations and inequalities

In the Equations window we can solve:

••equations

••inequalities

••simultaneous equations

••simultaneous inequalities

••inequalities linked with OR


Graphs of functions

The program can help to draw the graphs of three kinds of function.

•Graphs of linear functions

•Graphs of linear functions from coefficients

•Graphs of quadratic functions

•Graphs of linear quotient functions -- basic level and advanced level


Marking sets.
Marking sets

•Marking points on the coordinate diagram

•Intervals and inequality signs

•Linear inequalities with two unknowns

•Simultaneous inequalities with one unknown

•Simultaneous linear inequalities with two unknowns

•Inequalities linked with OR


Equations and inequalities can be written more simply. Intervals and inequalities can be written more simply.
Equations and inequalities can be written more simply

The messages "Intervals and inequalities can be written more simply", "Equations and inequalities can be written more simply" appear when one of the conditions used is redundant.

Example 1

• x<-4 AND x≥2  ⇔  contradiction

We can see this easily using the window for marking sets.

Example 2

• (x≤ 2  OR x≥3) AND x<5   ⇔  x≤ 2 OR (x≥3 AND x<5)

Example 3

• x∈(2;3] AND y≥3 AND y∈[5;6]   ⇔  x∈(2;3] AND y∈[5;6]


The point does not belong to the straight line
The point does not belong to the straight line.

The graph of a linear function   Graphs of linear functions from coefficients

The message "The point does not belong to the straight line" appears when we calculate the value of the function for the given argument incorrectly.

Example

• y=2x+1

     for the argument  x=0  the value of the function is   y(x)=2x+1=2•0+1=1

     for the argument  x=1  the value of the function is   y(x)=2x+1=2•1+1=3

     for the argument  x=-2  the value of the function is   y(x)=2x+1=2•(-2)+1=-3

     etc

If we have specified in the table that for  x=0  the value of the function is  y=3

the computer will show the above message, as the correct values are x=0, y=1.


The point does not belong to the hyperbola
The point does not belong to the hyperbola.

The graph of a linear quotient function -- basic level and advanced level

The message "The point does not belong to the hyperbola" appears when we calculate the value of the function for the given argument incorrectly.

•Example

for the argument  x=6  the value of the function is
for the argument  x=1  the value of the function is
for the argument  x=-2  the value of the function is
     etc

If we have specified in the table that for  x=1  the value of the function is  y=3

the computer will show the above message, as the correct values are x=1, y=6.


The point does not belong to the parabola
The point does not belong to the parabola.

The graph of a quadratic function

The message "The point does not belong to the parabola" appears when we calculate the value of the function for the given argument incorrectly.

•Example 1

     for the argument  x=-1  the value of the function is   y(x)=3•(-1)•(-1)=3

     for the argument  x=0  the value of the function is   y(x)=3•0•0=0

     for the argument  x=1 the value of the function is   y(x)=3•1•1=3

     etc

If we have specified in the table that for  x=-1  the value of the function is  y=-3

the computer will show the above message, as the correct values are x=-1, y=3.

•Example 2

     for the argument  x=0  the value of the function is   y(x)=0•0-3•0+1=3

     for the argument  x=1  the value of the function is   y(x)=1•1-3•1+1=-1

     for the argument  x=-2  the value of the function is   y(x)=(-2)•(-2)-3•(-2)+1=11

     etc

If we have specified in the table that for  x=1  the value of the function is  y=3

the computer will show the above message, as the correct values are x=1, y=-1.


Wrong regions marked - number line
Wrong regions marked.

Simultaneous inequalities with one unknown   Inequalities linked with OR

The message "Wrong regions marked" appears when we have marked points or line segments incorrectly.

•Example

	This is the correctly marked region.

If we mark regions in another way, eg.

	or
the computer will show the message "Wrong regions marked".


Wrong regions marked - rectangular coordinates
Wrong regions marked.

Simultaneous linear inequalities with two unknowns

The message "Wrong regions marked" appears when we have marked points, line segments or regions incorrectly.

•Example 1

	This is the correctly marked region.

If we mark regions in another way, eg.

the computer will show the message "Wrong regions marked".

NB. If there is a point, line or interval belonging to the region we have to mark it separately.

•Example 2

This is the correctly marked region.

If we only mark regions

the computer will show the message "Wrong regions marked".


Error in a function definition
Error in a function definition.

Linear functions - definitions   Quadratic functions - definitions   Linear quotient functions - definitions

Using 2xy we can draw

striaght lines
hyperbolas
parabolas

When we write the function definition we have to remember to

•have performed reduction of similar terms and cancellation of fractions, eg.

	y=2x+1+3x-4	write as	y=5x-3
		write as
•take a variable of the given exponent outside a bracket, if needed

		write as
		write as
		write as
•order the terms in the function definition from the highest power of the variable to the lowest

		write as
•always write the hyperbola definition as the quotient of linear functions

		write as
		write as


Write the right part of the condition as a linear function
Write the right part of the condition as a linear function.

When we write the linear function definition (y=ax+b) we have to remember to

•have performed reduction of similar terms and cancellation of fractions, eg.

    y=2x+1+3x-4  write as  y=5x-3

•take a variable of the given exponent outside a bracket, if needed

		write as
•order the terms in the function definition from the highest power of the variable to the lowest

    y=4+2x  write as  y=2x+4.


Error in a hypebola definition
Error in a hyperbola definition.

When we write the hyperbola definition we have to remember to

•have performed reduction of similar terms and cancellation of fractions, eg.

     2x+1+3x-4   write as   5x-3

•take a variable of the given exponent outside a bracket, if needed

		write as
•order the terms in the function definition from the highest power of the variable to the lowest

     1+3x   write as   3x+1

•always write the hyperbola definition as the quotient of linear functions

		write as
•move the minus sign into the numerator, if needed

		write as


Arguments and values of the function
Arguments and values of the function.

If we have a function of the form  y=f(x)  then we call  y - the value of the function f, and  x - the argument of the function  f

If we want to draw a few functions in the same window we have to remember to use the same variables to denote the arguments and the values of each function, eg.

we draw the functions  y=2x+1,  y=3x-6  (y - value of the function,  x - argument)

•if we write as a next function to draw  x=-2y  (x - value of the function,  y - argument) the computer will show the message "Decide which is the argument of`the function and which is its value!" (the correct notation is y=-0.5x )

•if we write as a next function to draw  z=4m+1 (z - value of the function,  m - argument) the computer will show the message "The variables must be marked by the same letters for all functions!". To draw a graph we have to change the names of the variables for y, x  or open a new graphs of functions window .


Errors and messages

•Equations and inequalities can be written more simply.

•Intervals and inequalities can be written more simply.

•Ask for a hint elsewhere.

•Too large numbers.

•Too many operations.

•The symbols cannot be placed together.

•This symbol may not appear in this position.

•It was much better.

•Something is missing.

•Wrong region markerd.

•Error in the solution.

•Increasing and reducing the number of solutions.

•Expansion and restriction of the domain.

•Error in the domain definition.

•Error when operating on both sides.

•The point does not belong to the straight line.

•The point does not belong to the hyperbola.

•The point does not belong to the parabola.

•Wrong regions marked - number line.

•Wrong regions marked - rectangular coordinates.

•Error in a function definition.


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