One approach: plug in the two boundaries from the graph (-5 and 3) into each answer choice. In the wrong answers, at least one of the numbers will result in an invalid statement. In the right answer, both of these numbers will result in a valid equality. For instance plugging -5 and 3 into |x+1|<=4 will result in 4=4. This is the right answer.

The full solution below is taken from the GMATFix App.



-Patrick

BlueDragon2010 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

|x-y| = the DISTANCE between x and y

The midpoint of the range = -1.
x can be any value from -5 (4 places BELOW -1) to 3 (4 places ABOVE -1).
In other words: the distance between x and -1 is less than or equal to 4.
In math terms:
|x-(-1)| ≤ 4.
|x+1| ≤ 4.

The correct answer is E.

An alternate approach is to plug numbers into the answer choices.
The correct answer choice must work for EVERY value between -5 and 3, inclusive -- and for no values outside this range.
Since |x| = the distance between x and 0, start with the value furthest from 0:
Let x=-5.
Eliminate any answer choice in which x=-5 doesn't work.

Answer choice A: |x|≤3
|-5|≤3.
5≤3.
Doesn't work. Eliminate A.

Answer choice B: |x|≤5
|-5|≤5
5≤5.
This works. Hold onto B.

Answer choice C: |x-2|≤3
|-5-2|≤3
7≤3.
Doesn't work. Eliminate C.

Answer choice D: |x-1|≤4
|-5-1|≤4
6≤4.
Doesn't work. Eliminate D.

Answer choice E: |x+1|≤4
|-5+1|≤4
4≤4.
This works. Hold onto E.

Only B and E remain.
Answer choice B -- |x|≤5 -- includes every value from -5 to 5, but the needed range is only from -5 to 3.
Eliminate B.

The correct answer is E.