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4.5 Solving Absolute Value Inequalities 10/30/13

4.5 Solving Absolute Value Inequalities 10/30/13

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Page 1: 4.5 Solving Absolute Value Inequalities 10/30/13

4.5 Solving Absolute ValueInequalities

10/30/13

Page 2: 4.5 Solving Absolute Value Inequalities 10/30/13

or

Solve an “Or” Compound Inequality

Solve or

From 4.1: Compound Inequlalities

Page 3: 4.5 Solving Absolute Value Inequalities 10/30/13

Solve an “And” Compound Inequality

Solve

SOLUTION

Page 4: 4.5 Solving Absolute Value Inequalities 10/30/13

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

Page 5: 4.5 Solving Absolute Value Inequalities 10/30/13

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

Page 6: 4.5 Solving Absolute Value Inequalities 10/30/13

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

Page 7: 4.5 Solving Absolute Value Inequalities 10/30/13

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

Page 8: 4.5 Solving Absolute Value Inequalities 10/30/13

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

“AND”

“OR”

Original Problem Solutions: Graph of solution|x| < 3 -3 < x < 3 Shade between

the 2 numbers|x| ≤ 3 -3 ≤ x ≤ 3

|x| > 4 x > 4 or x < -4

|x| ≥ 4 x ≥ 4 or x ≤ -4

Shade opposite direction (away from each other) from 4 and -4Shade opposite

direction (away from each other)

Page 9: 4.5 Solving Absolute Value Inequalities 10/30/13

|𝑎𝑥+𝑏|<𝑐

|𝑎𝑥+𝑏|≤𝑐

−𝑐<𝑎𝑥+𝑏<𝑐

−𝑐 ≤𝑎𝑥+𝑏≤𝑐

Solving Absolute Value Inequalities

Original Problem Rewrite as Graph of solution

|𝑎𝑥+𝑏|>𝑐 or

|𝑎𝑥+𝑏|≥𝑐 or

Page 10: 4.5 Solving Absolute Value Inequalities 10/30/13

Example 1 Solve an Inequality of the Form

Solve . Then graph the solution.

+x b ≤ c

+x 4 ≤ 10

– +x 4 ≤ 10 Write as “AND” compound inequality.10 ≤

– x ≤ 614 ≤

16–.14– 12– 10– 8– 6– 4– 2– 0 2 4 6 8

.

+16 4– 10≤ +0 4 10≤ +8 4 10≤

12 10≤ 4 10≤ 12 10≤

CHECK Test one value from each region of the graph.

SOLUTION

-4 - 4 -4

Page 11: 4.5 Solving Absolute Value Inequalities 10/30/13

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

Page 12: 4.5 Solving Absolute Value Inequalities 10/30/13

Example 2 Solve an Inequality of the Form

Solve . Then graph the solution.

+ax b c<

+2x 3 7<

– Write as “AND” compound inequality.+2x 3 7<7 <

–10 2x 4<<

–5 x 2<<

6– 5– 4– 3– 0 1 22– 1– 3

SOLUTION

-3 - 3 -3

2 2 2

Page 13: 4.5 Solving Absolute Value Inequalities 10/30/13

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Absolute Value Inequality

2. 1 4x 3<+ANSWER

1– 0..

– x 0.51 < <

13.

Page 14: 4.5 Solving Absolute Value Inequalities 10/30/13

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≥–

Rewrite as “OR” Compound Inequalities

+1 +1

+1 +1

8

Page 15: 4.5 Solving Absolute Value Inequalities 10/30/13

Checkpoint

Solve the inequality. Then graph your solution.

Solve an Absolute Value Inequality

4. 2x 3 7+ >

5. 4x 1 5+ ≥

ANSWER –1.5 orx x 1≥

1– 0 12–. .

ANSWER < –5 orx 2>x

4– 2– 0 2 4 66–

5–

Page 16: 4.5 Solving Absolute Value Inequalities 10/30/13

Homework: WS 4.5

Quiz 4.3-4.5 Friday

“I tried to catch some fog. I mist.”