Name the property: If a > b, then a + c > b + c.Name the property: If a > b, then a + c > b + c.

Addition Property of Inequality

Addition Property of Inequality

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If a > b and c > 0, then ac ___ bc.If a > b and c > 0, then ac ___ bc.

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If a > b and c < 0, then ac ___ bc.If a > b and c < 0, then ac ___ bc.

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Solve x + 3 < 5.Solve x + 3 < 5.

x < 2x < 2

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Solve – 2x > 8.Solve – 2x > 8.

x < – 4x < – 4

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Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b, then a + c < b + c.

6 < 7 and 6 + 3 < 7 + 3; i.e., 9 < 10

Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b, then a – c < b – c.

– 4 < – 2 and – 4 – 5 < – 2 – 5; i.e., – 9 < – 7

Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b and c > 0, then ac < bc.

2 < 5 and 2(3) < 5(3); i.e., 6 < 15

Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b and c < 0, then ac > bc.

2 < 5 and 2(– 3) < 5(– 3); i.e., – 6 > – 15

Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b and c > 0, then < .If a < b and c > 0, then < .a

cac

bcbc

4 < 8 and < ; i.e., 2 < 4

4 < 8 and < ; i.e., 2 < 4

4242

8282

Properties of InequalitiesProperties of Inequalities

PropertyProperty

ExampleExample

If a < b and c < 0, then > .If a < b and c < 0, then > .a

cac

bcbc

4 < 8 and ; i.e., – 2 > – 4

4 < 8 and ; i.e., – 2 > – 4

4– 24

– 28

– 28

– 2>>

Reverse signs if:Reverse signs if:

• multiplying by a negative• multiplying by a negativex4x4

– > 10 x < – 40– > 10 x < – 40

• dividing by a negative• dividing by a negative

– 3x < 6 x > – 2– 3x < 6 x > – 2

Solve – 4x + 3 > 23, and graph the solution.Solve – 4x + 3 > 23, and graph the solution.

x < – 5x < – 5

– 4x + 3 – 3 > 23 – 3– 4x + 3 – 3 > 23 – 3– 4x > 20– 4– 4 – 4– 4

00– 1– 1– 2– 2– 3– 3– 4– 4– 5– 5– 6– 6

Example 1Example 1

Solve – x + 8 ≤ 23, and graph the solution.Solve – x + 8 ≤ 23, and graph the solution.

x ≥ – 20x ≥ – 20

3434

– x + 8 – 8 ≤ 23 – 8– x + 8 – 8 ≤ 23 – 83434

– x ≤ 15– x ≤ 153434

4343

–– 4343

––55

Example 2Example 2

101000– 10– 10– 20– 20– 30– 30

Solve – x + 8 ≤ 23, and graph the solution.Solve – x + 8 ≤ 23, and graph the solution.

3434

x ≥ – 20x ≥ – 20

Example 2Example 2

– 2r > 27– 2r > 27

Solve – 2(r + 4) > 19, and graph the solution.Solve – 2(r + 4) > 19, and graph the solution.

r < – 13.5r < – 13.5

– 2r – 8 > 19– 2r – 8 > 19

– 2– 2– 2– 2

– 2r – 8 + 8 > 19 + 8– 2r – 8 + 8 > 19 + 8

Example 3Example 3

r < – 13.5r < – 13.5

– 12– 12– 13– 13– 14– 14– 15– 15– 16– 16

Solve – 2(r + 4) > 19, and graph the solution.Solve – 2(r + 4) > 19, and graph the solution.

Example 3Example 3

Solve 2x ≥ – 2x + 8.Solve 2x ≥ – 2x + 8.

x ≥ 2x ≥ 244 44

2x + 2x ≥ – 2x + 2x + 82x + 2x ≥ – 2x + 2x + 84x ≥ 84x ≥ 8

Example 4Example 4

Solve 3x – 1 < x + 9.Solve 3x – 1 < x + 9.

x < 5x < 522 22

3x – 1 – x < x + 9 – x3x – 1 – x < x + 9 – x2x – 1 < 92x – 1 < 9

2x – 1 + 1 < 9 + 12x – 1 + 1 < 9 + 12x < 102x < 10

Example 5Example 5

Solve 5x – 8 ≤ 9x + 2.Solve 5x – 8 ≤ 9x + 2.

x ≥ – 2.5x ≥ – 2.5– 4– 4 – 4– 4

5x – 8 – 9x ≤ 9x + 2 – 9x5x – 8 – 9x ≤ 9x + 2 – 9x– 4x – 8 ≤ 2– 4x – 8 ≤ 2

– 4x – 8 + 8 ≤ 2 + 8– 4x – 8 + 8 ≤ 2 + 8– 4x ≤ 10– 4x ≤ 10

Example 6Example 6

Solve 5x + 3 < – 7.Solve 5x + 3 < – 7.

x < – 2x < – 2

ExampleExample

Solve – 3x + 5 > 17.Solve – 3x + 5 > 17.

x < – 4x < – 4

ExampleExample

Solve 2x – 12 < 7x + 13.Solve 2x – 12 < 7x + 13.

x > – 5x > – 5

ExampleExample

Solve 2(n + 7) > – 3n + 12.Solve 2(n + 7) > – 3n + 12.

n > – n > – 2525

ExampleExample

Solve 3(y – 12) < – 2(y – 9) + 1.Solve 3(y – 12) < – 2(y – 9) + 1.

y < 11y < 11

ExampleExample

Solve 2.5(x – 3) – 3(x – 2.5) > 2x.Solve 2.5(x – 3) – 3(x – 2.5) > 2x.

x < 0x < 0

ExampleExample

Solve 8(0.75x – 0.375) < 12(1.25x + 0.5).Solve 8(0.75x – 0.375) < 12(1.25x + 0.5).

x > – 1x > – 1

ExampleExample

Solve + 3 ≥ 12.Solve + 3 ≥ 12.

b ≥ 45b ≥ 45

b5b5

ExampleExample

Solve ≥ 12.Solve ≥ 12.

b ≥ 57b ≥ 57

b + 35

b + 35

ExampleExample

Solve ≤ .Solve ≤ .

a ≥ 39a ≥ 39

a + 37

a + 37

a – 95

a – 95

ExampleExample

Using the inequality ax – b ≤ c, and assuming that a, b, and c are real numbers with a ≠ 0, solve the inequality for x. Be careful to account for all possible values of a, b, and c.

Using the inequality ax – b ≤ c, and assuming that a, b, and c are real numbers with a ≠ 0, solve the inequality for x. Be careful to account for all possible values of a, b, and c.

ExerciseExercise

Using the inequality + s > t,

and assuming that r, s, and t are real numbers, solve the inequality for x. Be careful to account for all possible values of r, s, and t.

Using the inequality + s > t,

and assuming that r, s, and t are real numbers, solve the inequality for x. Be careful to account for all possible values of r, s, and t.

xrxr

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