Answer:
b. –13 < x < 23
Explanation:
You want the solution to the inequality |x -5| +2 < 20.
Solution
|x -5| < 18 . . . . . . . subtract 2
-18 < x -5 < 18 . . . . . unfold
-13 < x < 23 . . . . . . . . . add 5
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Additional comment
The absolute value function is piecewise defined:
|x| = x . . . for x ≥ 0
|x| = -x . . . for x < 0
Technically, this means we need to write the given inequality as two inequalities, each with its own domain:
x - 5 + 2 < 20 . . . . for x ≥ 5
-(x -5) +2 < 20 . . . . for x < 5
Then the solution is in two parts:
-13 < x . . . for x < 5
x < 23 . . . for x ≥ 5
The union of these two solution sets is the compound inequality shown above.
Note that if the original inequality symbol is reversed (> 20), then this method of decomposing the absolute value inequality must be used, and the final solution set will have two disjoint parts.
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