Official Guide Explanation:
Problem Solving #130




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Solution and Metadata

Question: 130
Page: 170
Difficulty: 5 (Moderate)
Category 1: Geometry > Lines >
Category 2: Algebra > Inequalities > other
Category 3: Algebra > Absolute Value >

Explanation: Recognize that any equation that includes absolute value symbols implies two separate equations: one when the expression inside the absolute value signs is positive, one when it's negative. The number line indicates that you're looking for the expression that gives you the two equations x ≤ 3 and x ≥ -5. (A) and (B) can be eliminated, because those will give you 3 and -3 and 5 and -5 as the endpoints. Try the remaining choices:

(C): positive: x - 2 ≤ 3, or x ≤ 5. That's not one of the ones we need, so we can stop there.

(D): positive: x - 1 ≤ 4, or x ≤ 5. Same problem. At this point, you probably don't need to check (E) to know that it's correct, but I'll show the solution for the sake of completeness.

(E): positive: x + 1 ≤ 4, or x ≤ 3

    negative: -(x + 1) ≤ 4, or x + 1 ≥ -4, or x ≥ -5

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