Walkabout wrote:

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Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

(A) |x| <= 3

(B) |x| <= 5

(C) |x - 2| <= 3

(D) |x - 1| <= 4

(E) |x +1| <= 4

We start by expressing the interval on the number line as an inequality:

-5 ≤ x ≤ 3

Looking at answer choices A and B, we see that those two equations will not produce the inequality shown above. Thus, we consider answer choices C, D, and E.

When we solve an absolute-value equation with one absolute-value expression, we consider two cases: one with the positive version of the expression inside the absolute value bars and one with the negative (or opposite) version of the expression inside the absolute value bars. Let’s use this fact to evaluate answer choice C:

Answer choice C: |x - 2| ≤ 3

Case 1: Expression Positive:

x – 2 ≤ 3

x ≤ 5

Case 2: Expression Negative:

-(x - 2) ≤ 3

-x + 2 ≤ 3

-x ≤ 1

x ≥ -1

The solution is x ≤ 5 and x ≥ -1, i.e., -1 ≤ x ≤ 5. However, this does not fit the interval represented on the number line.

Answer choice D: |x - 1| ≤ 4

Case 1: Expression Positive:

x - 1 ≤ 4

x ≤ 5

Case 2: Expression Negative:

-(x – 1) ≤ 4

-x + 1 ≤ 4

-x ≤ 3

x ≥ -3

The solution is x ≤ 5 and x ≥ -3, i.e., -3 ≤ x ≤ 5. This does not fit the interval represented on the number line.

Answer Choice E: |x +1| ≤ 4

Case 1: Expression Positive:

x + 1 ≤ 4

x ≤ 3

Case 2: Expression Negative:

-(x + 1) ≤ 4

-x – 1 ≤ 4

-x ≤ 5

x ≥ -5

The solution is x ≤ 3 and x ≥ -5, i.e., -5 ≤ x ≤ 3. This DOES describe the interval represented on the number line.

Answer E

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