This document provides instruction on solving absolute value equations and inequalities. It begins by explaining what absolute value equations like |x|=5 mean and the two cases to consider when solving. Examples are provided of solving equations like |x+4|=5 and |x-7|=8. The document also explains how to write absolute value equations and solve inequalities, providing examples like solving |3y-3|>9 for the values of y. Key steps are outlined for both absolute value equations and inequalities.

1. 6-5 Solving Open Sentences
Involving Absolute Value
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2. Objectives
• Students will solve absolute value equations.
• Students will solve absolute value
inequalities.
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3. 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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4. Example 1
|x + 4| = 5
Case 1: x + 4 = 5
x=1
Case 2: x + 4 = -5
x = -9
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5. Example 2
|x – 7| = 8
Case 1: x – 7 = 8
x = 15
Case 2: x – 7 = -8
x = -1
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6. 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. distance from midpoint
to endpoints
•
• |x - 6| = 3
3 9
midpoint
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7. 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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8. Example 3
|3y – 3| > 9
3y – 3 > 9 or 3y – 3 < -9
3y > 12 or 3y < -6
y>4 or y < -2
-2 4
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9. Example 4
|s - 3| ≤ 12
s – 3 ≤ 12 and s – 3 ≥ -12
s ≤ 15 and s ≥ -9
-9 15
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