Which of the following inequalities is equivalent to –2 < x < 4 ?

(A) | x – 2 | < 4 => -2<x<6: out
(B) | x – 1 | < 3 => –2 < x < 4: yeah
(C) | x + 1 | < 3 => -4<x<2: out
(D) | x + 2 | < 4 => -6<x<2: out
(E) None of the above: out

Ans: B

The only golden rule needed to solve this excellent question => |x|<n => x lies in the range (-x,x)

There is a super easy method to solve such question
First remove the mod
then set the expression to 0
then get a value of X
then add and subtract the value from the given number on the RHS

Lets try with an example (B) --------------> | x – 1 | < 3
First remove the mod -------------------> x-1
then set the expression to 0--------------> x-1=0
then get a value of x---------------------> x=1
then add and subtract the value from the given number on the RHS 1+3=4 and 1-3= -2
the range become -2 to 4 correct answer

Now try its with an incorrect option (C) | ----------> |x + 1 | < 3
First remove the mod --------------------------> x+1
then set the expression to 0----------------------> x+1=0
then get a value of x------------------------------> x=-1
then add and subtract the value from the given number on the RHS -1+3=2 and -1-3= -4
the range become -4 to 2 Incorrect

THE CORRECT ANSWER IS B

The best way to do this question would be through number line.

Option b says distance of point x from 1 should be less than 3 which means from 1 you can move to the right till 4 and from 1 you can move left till -2 so that the distance from 1 stays less than 3.
-2<x<4.

Note: | x – 3 | < 1 represents the distance from x to 3 is less than 1. On a number line, 3 is the mid-point, x is within the range of one unit to the left of 3 and one unit to the right of 3. Therefore, 2<x<4.

Question ask: Which of the following inequalities is equivalent to –2 < x < 4 ?

We can reverse engineer the equation. From –2 < x < 4, we can find the mid-point is 1, and x is within the range of 3 units to the left and 3 units to the right. Then, it is quite straightforward to see the answer is B. | x – 1 | < 3