Jeevanantham wrote: -5 -4 -3 -2 -1 0 1 2 3

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

A) |x| <= 3

B) |x| <= 5

C) |x-2| <= 3

D) |x-1| <= 4

E) |x+1| <= 4

Please explain.

|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.