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6-5 Solving Open Sentences

Involving Absolute Value

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Objectives

• Students will solve absolute value equations.

• Students will solve absolute value

inequalities.

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Absolute Value Equations

|x|=5 means the distance between x and 0 is 5

units, so x = 5 or –5.

When solving equations, there are 2 cases to

consider:

1. The value inside the absolute value signs is

positive.

2. The value inside the absolute value signs is

negative.

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Example 1

|x + 4| = 5

Case 1: x + 4 = 5

x = 1

Case 2: x + 4 = -5

x = -9

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Example 2

|x – 7| = 8

Case 1: x – 7 = 8

x = 15

Case 2: x – 7 = -8

x = -1

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Writing Absolute Value Equations

• Find the point that is in the middle of the 2

points.

• Find the distance between the midpoint and

the endpoints.

|x - 6| = 3 3 9

midpoint

distance from midpoint

to endpoints•

•

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Solving Absolute Value

Inequalities

• Case 1: The value inside the absolute value

signs is greater than the given value of n.

• Case 2: The value inside the absolute value

signs is less than the opposite of n.

• If |x|<n, then x<n and x>-n “within” “and”

• If |x|>n, then x>n and x<-n “outside” “or”

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Example 3

|3y – 3| > 9

3y – 3 > 9 or 3y – 3 < -9

3y > 12 or 3y < -6

y > 4 or y < -2

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Example 4

|s - 3| ≤ 12

s – 3 ≤ 12 and s – 3 ≥ -12

s ≤ 15 and s ≥ -9

-9 15