2.4 – Linear Inequalities in One Variable

An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥.

Equations Inequalities

x = 3

12 = 7 – 3y

x > 3

12 ≤ 7 – 3y

A solution of an inequality is a value of the variable that makes the inequality a true statement.

The solution set of an inequality is the set of all solutions.

2.4 – Linear Inequalities in One Variable

2.4 – Linear Inequalities in One Variable

Example

Graph each set on a number line and then write it in interval notation.

a. { | 2}

b. { | 1}

c. { | 0.5 3}

x x

x x

x x

a. [2, )

b.

c. (0.5,3]

2.4 – Linear Inequalities in One Variable

Addition Property of Inequality

If a, b, and c are real numbers, then

a < b and a + c < b + c

a > b and a + c > b + c

are equivalent inequalities.

Also,

If a, b, and c are real numbers, then

a < b and a - c < b - c

a > b and a - c > b - c

are equivalent inequalities.

2.4 – Linear Inequalities in One Variable

ExampleSolve: Graph the solution set.3 4 2 6x x

{ | 10} or 10,x x

[

2.4 – Linear Inequalities in One Variable

Multiplication Property of Inequality

If a, b, and c are real numbers, and c is positive, thena < b and ac < bc are equivalent inequalities.

If a, b, and c are real numbers, and c is negative, thena < b and ac > bc

are equivalent inequalities.

2.4 – Linear Inequalities in One Variable

The direction of the inequality sign must change when multiplying or dividing by a negative value.

Solve: Graph the solution set.

{ | 3} or 3,x x

2.3 6.9x

The inequality symbol is reversed since we divided by a negative number.

(

Example

2.4 – Linear Inequalities in One Variable

Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.

3x + 9 ≥ 5x – 53x – 3x + 9 ≥ 5x – 3x – 5

9 ≥ 2x – 5

14 ≥ 2x7 ≥ x

9 + 5 ≥ 2x – 5 + 5

3x + 9 ≥ 5(x – 1)

x ≤ 7

[

2.4 – Linear Inequalities in One Variable

ExampleSolve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set.

7(x – 2) + x > –4(5 – x) – 12

7x – 14 + x > –20 + 4x – 12

8x – 14 > 4x – 32

8x – 4x – 14 > 4x – 4x – 32

4x – 14 > –32

4x – 14 + 14 > –32 + 14

4x > –18

x > –4.5(

2.4 – Linear Inequalities in One Variable

Intersection of Sets

The solution set of a compound inequality formed with and is the intersection of the individual solution sets.

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Example

Find the intersection of: {2,4,6,8} {3,4,5,6}

The numbers 4 and 6 are in both sets.

The intersection is {4, 6}.

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Solve and graph the solution for x + 4 > 0 and 4x > 0.

Example

First, solve each inequality separately.

x + 4 > 0

x > – 4

4x > 0

x > 0and

-4(

0(

( (0, )

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Example

0 4(5 – x) < 8

0 20 – 4x < 8

0 – 20 20 – 20 – 4x < 8 – 20

– 20 – 4x < – 12

5 x > 3

Remember that the sign direction changes when you divide by a number < 0!

(

[

(3,5]

2.4 – Linear Inequalities in One Variable

3 4 5

Compound Inequalities

Example – Alternate Method

0 4(5 – x)

0 20 – 4x

0 – 20 20 – 20 – 4x

– 20 – 4x

5 x (

[

(3,5]

2.4 – Linear Inequalities in One Variable

3 4 5

4(5 – x) < 8

20 – 4x < 8

20 – 20 – 4x < 8 – 20

– 4x < – 12

x > 3

0 4(5 – x) < 8

Dividing by negative:

change sign

Dividing by negative:

change sign

Compound Inequalities

The solution set of a compound inequality formed with or is the union of the individual solution sets.

Union of Sets

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Find the union of:

Example

{2,4,6,8} {3,4,5,6}

The numbers that are in either set are {2, 3, 4, 5, 6, 8}.

This set is the union.

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Example: Solve and graph the solution for 5(x – 1) –5 or 5 – x < 11

5(x – 1) –5

5x – 5 –5

5x 0

x 0

5 – x < 11

–x < 6

x > – 6

or

0[

-6(

(–6, )-6(

2.4 – Linear Inequalities in One Variable

Compound Inequalities

Example:

or

,

2.4 – Linear Inequalities in One Variable

Compound Inequalities

2.4 – Linear Inequalities in One Variable