Question

The number of integral value(s) of $x,$ satisfying the inequality $2|x+2|-|x+5|\le 4$ is

• A. $0$
• B. $5$
• C. $10$
• D. $7$

Answer 1

The correct option is C $10$

$2|x+2|-|x+5|\le 4$

Case $1:$ If $x<-5$
then $-2x-4+x+5\le 4$
$\Rightarrow -x+1\le 4$
$\Rightarrow x\ge -3$, which is not possible.

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

Case $3:$ If $x\ge -2$
then $2x+4-x-5\le 4$
$\Rightarrow x-1\le 4$
$\Rightarrow x\le 5$
$\therefore x\in [-2,5]$

From all the above cases, we can conclude that
$x\in [\frac{-13}{3},-2)\cup [-2,5]$
i.e., $x\in [\frac{-13}{3},5]$
Integral values of $x$ are $-4,-3,-2,-1,0,1,2,3,4,5$
Number of integral values of $x$ is $10$