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.
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Algebra Solving Open Sentences Involving Absolute Value
1. 6-5 Solving Open Sentences
Involving Absolute Value
This presentation was created following the Fair Use
Guidelines for Educational Multimedia. Certain materials are
included under the Fair Use exemption of the U. S. Copyright
Law. Further use of these materials and this presentation is
restricted.
1
2. Objectives
• Students will solve absolute value equations.
• Students will solve absolute value
inequalities.
2
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.
3
4. Example 1
|x + 4| = 5
Case 1: x + 4 = 5
x=1
Case 2: x + 4 = -5
x = -9
4
5. Example 2
|x – 7| = 8
Case 1: x – 7 = 8
x = 15
Case 2: x – 7 = -8
x = -1
5
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
6
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”
7
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
8
9. Example 4
|s - 3| ≤ 12
s – 3 ≤ 12 and s – 3 ≥ -12
s ≤ 15 and s ≥ -9
-9 15
9
