Chapter 6: Solving Linear Inequalities
6.6 – Solving Inequalities Involving Absolute Values
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Warm‐up:
Solve each inequality. Put solution set in interval notation.
1) p – 3 < 2
2) z + 5 ≤ 11
3) ‐4r > 16
4) ‐4r < ‐16
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6.6 Solving Inequalities Involving Absolute Value
• Objective
‐Be able to…
• Solve and graph absolute value inequalities.
• Write absolute value inequalities.
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Consider the inequality . This means that the distance between 0 to x is less than 5 units. Therefore, x > ‐5 and x < 5
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• When absolute value is on the left of the inequality…
• If it is a less than symbol, < or ≤, the compound sentence uses and.
‐ If absolute value is less than (or equal to) a negative number, there is no solution ( ).
• If it is a greater than symbol, > or ≥, the compound sentence uses or. ‐ If absolute value is greater than (or equal to) a negative number, the solution is all real numbers ( ).
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Key Concept
Solving Absolute Value Inequalities
When solving inequalities that involve absolute value, there are 2 cases to consider
Case 1: Case 2: Positive Case
Negative Case
Remove the absolute value sign and leave everything the same
• Remove the absolute value sign
• Flip the sign!!
• Change the right side to a negative 7
• Example 1: Solve an Absolute Value Inequality (<)
Solve each open sentence. Then graph the solution set.
1) Case 1: Case 2: 8
• Example 1: Solve an Absolute Value Inequality (<)
Solve each open sentence. Then graph the solution set.
2) Case 1: Case 2: 9
• Example 2: Solve an Absolute Value Inequality (>)
Solve each open sentence. Then graph the solution set.
3) Case 1: Case 2: 10
• Example 2: Solve an Absolute Value Inequality (>)
Solve each open sentence. Then graph the solution set.
4) Case 1: Case 2: 11
Writing an open sentence involving absolute value from a graph
General Form:
OR
Where m = midpoint (equal distance from the 2 points)
and
d = distance from endpoint to midpoint
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• Example 1: Write an Absolute Value Inequality (<)
Write an open sentence involving absolute value for each graph.
5)
6)
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• Example 2: Write an Absolute Value Inequality (>)
Write an open sentence involving absolute value for each graph.
7)
8)
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