INEQUALITIES Brought To You By- Tutorial Services-The Math Center.

INEQUALITIES

Brought To You By-

Tutorial Services-The Math Center

Inequalities

An inequality is a comparison of values There are four possible relationships

between different values. Greater than, less than, greater than or equal

to, and less than or equal to. The four symbols used to represent these

relationships respectively are : >,<, ≥, ≤

Example: 3 < 5

In this case, the inequality states that 3 is less than 5.

Think about the symbol as being an arrow that always points toward the smaller value.

2 < 5 < 7

or

9 > 5 > 3

Inequalities also work with variables

Example:

x < 5 means that x can be any value that is less than 5.

x ≥ 3 means that x is any number greater than or equal to 3.

Graphing Inequalities in One Variable • The solution set for inequalities of one variable can be

graphed on a number line.

Example: x < 5x < 5 would be graphed as :

)

If x is less than or greater than 5, a parenthesis or an open circle is used in the graph because 5 is not included in the set.

x5-5

x5 -5

-OR-

Example: You would graph the solution of

3 ≤ x as follows

In this case, a bracket or a closed circle is used because 3 is included in the set.

x

53

x

53

-OR-

[

Some examples of graphing solutions

x

∞9

x

∞12

x

53

12 < x12 < x

3 ≤ x < 5

(

(

[ )

x > 9x > 9

Interval Notation

Another way of representing the solution set of inequalities is by using interval notation.

Example: The solution set of 3 ≤ x < 53 ≤ x < 5

can be written using interval notation as [3,5).

From this notation you can see that there is a left boundary and a right boundary which are 3 and 5.

However, if the number is in parenthesis, then it is not in the set but, numbers close to that boundary, such as 4.9 or 4.99, are included.

What happens if one of the boundaries is removed?

Example: x < 5x < 5

In this case, the values of x x go on forever in the negative direction.

We can represent this inequality in interval notation as (-∞,5), where the symbol (-∞), meaning negative infinity, is used to represent the fact that there is no endpoint for the values of XX in the negative direction.

x5-5

Is represented by a graph as :

)

Examples of interval notation:

x ≥ 3

x < 15

3 < x < 15

Answer: (-∞,15 )

Answer: [3,∞)

Answer: (3,15)

Solving Linear Inequalities of One Variable

Example: x+3 < 4x+3 < 4 is a linear inequality

All of the rules that apply to equations also apply to inequalities with one important exception.

If you ever multiply or divide by a negative number you must reverse the direction of the inequality sign.

Example: -2x+4 > 8

Subtract 4 from both sides: -2x+4 -4 >8 -4

Since we are dividing by a negative we have to change the direction of the inequality:

Answer: x < -2 or (-∞, 2)

Divide both sides by -2: -2x / -2 > 4 / -2

To solve for x :

X < -2

Example with X’s on both sides: 4x-6 > 2x+10

Add 6 to both sides: 4x -6 +6 > 2x+10 +6

Subtract 2x from both sides: 4x -2x > 2x+16 -2x

Answer: x>8 or (8,∞)

Divide both sides by 2: 2x /2 > 16 /2

Example: x+3 > 4x+6

Subtract 3 from both sides: x+3 -3 > 4x+6 -3

Subtract 4x from both sides: x -4x > 4x+3 -4x

Answer: x < -1 or (-∞,-1)

Divide both sides by -3: -3x / -3 > 3 / -3

Change the direction of the inequality since there is division by a negative number

Example: 5(x-2) ≥ 9x-3(2x-4)

Distributive Property:Distributive Property: 5x -10 ≥ 9x -6x +12

Add like terms: 5x-10 ≥ 3x+12

Answer: x ≥ 11 or [11,∞)

Add 10 to both sides: 5x-10 +10 ≥ 3x+12 +10

Divide both sides by 2: 2x / 2 ≥ 22 / 2

Subtract 3x from both sides: 5x -3x ≥ 3x +22 -3x

When you solve an inequality the solutions can be written in interval notation or set builder notation.

Set builder notation:Set builder notation:

and {x | 5 < x ≤ 9}

This notation says that x is any value such that it satisfies the inequality in the notation.

{x | x ≤ 5}

Solving Compound Inequalities Compound inequalities involve two inequalities of the same

variable.

Example: -3 < 2x+1 < 5 can be written as two inequalities

-3 < 2x+1 and 2x+1 < 5

Each inequality is solved separately. Another method to solve the compound inequality is to

apply properties of equations to all parts of the inequalities.

Example: -3 < 2x+1 < 5

Can be solved as a whole as follows:

Subtract 1 from all parts: -3 -1 < 2x +1 -1 < 5 -1

Answer: -2 < x < 2 or (-2,2)

or {x | -2 < x < 2}

Divide all parts by 2: -4 / 2 < 2x / 2 < 4 / 2

Solving Absolute Value Inequalities Solving inequalities involving absolute values requires

learning 2 new rules that will affect how the problem is solved.

1. When solving an inequality in which the absolute value expression is less than (<) or less than or equal to (≤) a standard expression, such as |x+3| ≤ 5, we must rewrite this as -5 ≤ x+3 ≤ 5 and solve.

2. When solving an inequality in which the absolute value expression is greater than (>) or greater than or equal to (≥) a standard expression, such as |2x-1| > 7, we must rewrite this as 2x-1 < -7 or 2x-1 > 7 and solve each inequality separately.

Example: |x+3| ≤ 5

Rewrite: -5 ≤ x+3 ≤ 5

Subtract 3 from all parts: -5 -3 ≤ x +3 -3 ≤ 5 -3

Simplifying gives answer(Note the answer is given

in three different ways) :

-8 ≤ x ≤ 2 or [-8,2] or {x | -8 ≤ x ≤ 2}

Example: |2x-1| > 7

Rewrite: 2x-1 < -7 or 2x-1 > 7

Add 1 to both sides of each inequality:

2x -1 +1 < -7 +1 or 2x-1 +1 > 7 +1

Simplifying gives the answer:

Divide both sides of each inequality by 2: 2x /2 < -6 /2 or 2x /2 > 8 /2

x < -3 or x > 4, (-∞,-3) U (4,∞) or {x | x < -3 or x > 4}

Example: |-2x+5| < 1

Rewrite: -1 < -2x +5 < 1

Subtract 5 from all parts -1 -5 < -2x +5 -5 < 1 -5

Answer: (2,3) or {x | 2 < x < 3}

Divide all parts by -2: -6 /-2 < -2x /-2 < -4 /-2

Change direction of inequality: 3 > x > 2

Links and handouts

Inequalities student handout Inequalities Quiz