Question

If $2|x+2|-|x+5|\le 4$, then $x\in$

• A. $[-2,5]$
• B. $(-\frac{13}{3},-2)$
• C. $[-5,5]$
• D. $[-\frac{13}{3},5]$

Answer 1

The correct option is D $[-\frac{13}{3},5]$

If $x<-5$,
$-2x-4+x+5\le 4$
$\Rightarrow -x+1\le 4$
$\Rightarrow x\ge -3$ Not possible

If $-5\le x<-2$,
$-2x-4-x-5\le 4\Rightarrow -3x-9\le 4\Rightarrow x\ge -\frac{13}{3}\Rightarrow x\in [-\frac{13}{3},-2)$

If $x\ge -2$,
$2x+4-x-5\le 4\Rightarrow x-1\le 4\Rightarrow x\le 5\Rightarrow x\in [-2,5]$

Hence, $x\in [-\frac{13}{3},-2)\cup [-2,5]$
i.e., $x\in [-\frac{13}{3},5]$