4.5 Solving Absolute ValueInequalities

11/16/12

Solve |x| < 3

What value of x would make what’s inside | | 7 or

-7 What numbers are less than 3 units away from 0?

Answer: Numbers to the right of -3, like -2, -1 and anything to the left of 3 like 2, 1

Mathematically written: -3 < x < 3

x > -3

x < 3

Solve |x| ≤ 3

Answer: Numbers to the right of -3 including 3 and anything to the left of 3 including 3.

Mathematically written: -3 ≤ x ≤ 3

x ≥ -3

x ≤ 3

Solve |x| > 4

What value of x would make what’s inside | | 7 or

-7 What numbers are more than 4 units away from 0?

Answer: Numbers to the right of 4, like 5, 6, 7 or anything to the left of - 4 like -5, -6, -7

Mathematically written: x < -4 or x > 4

x > 4

x < -4

Solve |x| ≥ 4

Answer: Anything to the right of 4 including 4 or anything to the left of - 4 including -4

Mathematically written: x ≤ -4 or x ≥ 4

x ≥ 4

x ≤ -4

Summary:Inequality Equivalent Form Graph

|x| < 3 -3 < x < 3 Shade between -3 and 3

|x| ≤ 3 -3 ≤ x ≤ 3

|x| > 4 x > 4 or x < -4 Shade opposite direction (away from each other) from 4 and -4|x| ≥ 4 x ≥ 4 or x ≤ -4

< and ≤ are both AND problems> and ≥ are both OR problems (think greater pronounced as greatOR)

“AND”

“OR”

Solving more complex Inequality

Example 1 Solve an Inequality of the Form

Solve . Then graph the solution.

+x b ≤ c

+x 4 ≤ 10

– +x 4 ≤ 10 Write equivalent compound inequality.10 ≤

– x ≤ 6 Subtract 4 from each expression.14 ≤

16–.14– 12– 10– 8– 6– 4– 2– 0 2 4 6 8

.

-4 -4 -4

SOLUTION

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Absolute Value Inequality

1. +x 1 ≤ 4

ANSWER – x ≤ 35 ≤

6– 4– 2– 0 2 4. .3

7 < x < 9ANSWER

Example 2 Solve an Inequality of the Form

Solve . Then graph the solution.

+ax b c<

+2x 3 7<

– Write equivalent compound inequality.+2x 3 7<7 <

– Subtract 3 from each expression.10 2x 4<<

– Divide each expression by 2.5 x 2<<

6– 5– 4– 3– 0 1 22– 1– 3

SOLUTION

-3 -3 -3

2 2 2

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Absolute Value Inequality

4. 2x 1 9<–

5. 1 4x 3<+ANSWER

1– 0..

– x 0.51 < <

1

ANSWER – x 54 < <

4– 2– 0 2 4

5

6

Example 3 Solve an Inequality of the Form

Solve . Then graph the solution.

+ax b c≥

x 1 32

1– ≥

FIRST INEQUALITY SECOND INEQUALITY

x 1 32

1– ≤ – x 1 3

2

1≥–

Add 1 to each side.x 2

2

1≤ – 4x

2

1≥

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Absolute Value Inequality

7. 2x 3 7+ >

8. 4x 1 5+ ≥

ANSWER –1.5 orx x 1≥

1– 0 12–. .

≤

ANSWER < –5 orx 2>x

4– 2– 0 2 4 66–

5–

Homework: 4.5 p.201 #11-20 all

22-28 even34-44 even