Question

The set $A=\left\{x:\left|2x+3\right|<7\right\}$ is equal to the set

• A. $A=\left\{x:0<x+5<7\right\}$
• B. $A=\left\{x:-3<x<7\right\}$
• C. $A=\left\{x:-7<x<7\right\}$
• D. $A=\left\{x:-13<2x<4\right\}$

Answer 1

The correct option is A

$A=\left\{x:0<x+5<7\right\}$

Explanation for correct option

Consider the inequality $\left|2x+3\right|<7$

Since the inequality has modulo sign, on simplifying,

$-7<2x+3<7$

$$\begin{gathered} \Rightarrow -7-3<2x<7-3 \\ \Rightarrow -10<2x<4 \\ \Rightarrow -5<x<2 \end{gathered}$$

On adding $5$ on all sides,

$$\begin{gathered} \Rightarrow -5+5<x+5<2+5 \\ \Rightarrow 0<x+5<7 \end{gathered}$$

Hence the correct option is option A.