BTGmoderatorDC wrote:

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

When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:
Rule #1: If |something| < k, then -k < something < k
Rule #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is positive

(A) |x| ≤ 3
Applying Rule #1, we get:
-3 ≤ x ≤ 3
This does not match the shaded part of the number line
ELIMINATE

(B) |x| ≤ 5
Applying Rule #1, we get:
-5 ≤ x ≤ 5
This does not match the shaded part of the number line
ELIMINATE

(C) |x - 2| ≤ 3
We get: -3 ≤ x - 2 ≤ 3
Add 2 to all parts of the inequality to get: -1 ≤ x ≤ 5
This does not match the shaded part of the number line
ELIMINATE

(D) |x - 1| ≤ 4
We get: -4 ≤ x - 1 ≤ 4
Add 1 to all parts of the inequality to get: -3 ≤ x ≤ 5
This does not match the shaded part of the number line
ELIMINATE

(E) |x +1| ≤ 4
We get: -4 ≤ x + 1 ≤ 4
Subtract 1 from all parts of the inequality to get: -5 ≤ x ≤ 3
PERFECT. This matches the shaded part of the number line

Answer: E

Cheers,
Brent