CAT 2017

Slot 2

QA

Question & Solution

$|f(x)+ g(x)| = |f(x)| + |g(x)|$ if and only if:

Case 1:
$f(x) \geq 0$ and $g(x) \geq 0$
<=> $2x-5 \geq 0$ and $7-2x \geq 0$
$\Rightarrow x \geq \frac{5}{2}$ and $\frac{7}{2} \geq x$ 
$\frac{5}{2}\leq x\leq\frac{7}{2}$

Case 2:
$f(x) \leq 0$ and $g(x) \leq 0$ $\Rightarrow 2x-5 \leq 0$ and $7-2x \leq 0$
$\Rightarrow x \leq \frac{5}{2}$ and $\frac{7}{2} \leq x$
 

$\Rightarrow$ So $x<=5/2$ and $x>=7/2$ which is not possible.

Hence, answer is $\boxed {\frac{5}{2}\leq x\leq\frac{7}{2}}$