What is solution set of the given equation and why?

Question

What is the solution set of the below given equation and why?

Find the solution set of the equation: $| 7 - x | < 2 , \forall x\in\mathbb R$

A) $\{ x \mid x\in\mathbb R, x < 5 \}$

B) $\{ x \mid x\in\mathbb R, 5 < x < 9 \}$

C) $\{ x \mid x\in\mathbb R, x < 9 \}$

D) $\{ x \mid x\in\mathbb R, -5 < x < 9 \}$

Answer 1

(A) $x=2, 5<2,$ which is absurd

(C)same absurdity as that of (A)

(D) same absurdity

(B) is the correct one.

Analytical method: $|7-x|=|x-7|<2 \implies -2<x-7<2 \implies 5<x<9.$

Answer 2

Everyone should know from high school that, if $b>0$, $$|x-a|<b \iff a-b<x<a+b.$$