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CAT 2017 [slot 2] Question with solution 32

Question 66:
Let $f(x) =2x-5$ and $g(x) =7-2x$. Then |f(x)+ g(x)| = |f(x)|+ |g(x)| if and only if
  1. $\frac{5}{5}<x<\frac{7}{2}$
  2. $x\leq\frac{5}{2}$ or $x\geq\frac{7}{2}$
  3. $x<\frac{5}{2}$ or $x\geq\frac{7}{2}$
  4. $\frac{5}{2}\leq x\leq\frac{7}{2}$
Option: 1
Explanation:

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

case 1: $f(x) \geq 0$ and $g(x) \geq 0$
<=> $ 2x-5 \geq 0 $ and $7-2x \geq 0$
<=> $ 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 $g(x) \leq 0$

<=> $ 2x-5 \leq 0 $ and $7-2x \leq 0$
<=> $ x \leq \frac{5}{2}$ and $ \frac{7}{2} \leq x$
So x<=5/2 and x>=7/2 which is not possible.

Hence, answer is

<=> $\frac{5}{2}\leq x\leq\frac{7}{2}$


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