Question

Solve $|x+3|-|x-4|\ge 4$

• A. $[\frac{5}{2},4)$
• B. $[-3,4]$
• C. $[\frac{5}{2},\infty )$
• D. $[4,\infty )$

Answer 1

The correct option is C $[\frac{5}{2},\infty )$

$|x+3|-|x-4|\ge 4$
If $x\ge 4$,
$x+3-x+4\ge 4$
$7\ge 4$ Always true.
$\Rightarrow x\in [4,\infty )$

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

If $x<-3$,
$-x-3+x-4\ge 4$
$\Rightarrow -7\ge 4$ Never true.
Hence, $x\in [\frac{5}{2},\infty )$