Question

The equation $|| x -1| +a| = 4$ can have real solutions for x if ‘a’ belongs to the interval

• A. $\left(-\infty, +\infty\right)$
• B. $(-\infty, 4]$
• C. $\left(4, +\infty\right)$
• D. $[-4, 4]$

Answer 1

Answer: (B)

$\left|x-1\right|+a\left|=4 \Rightarrow \right|\left|x-1\right|+a=\pm4$
$\Rightarrow \left|x-1\right|=-a \pm4$
The above equation holds if $-a+4 \ge0$ or $-a-4 \ge0$
$\Rightarrow a \le4$ or $a \le-4 \Rightarrow a \in(-\infty, -4]\cup (-\infty, -4]$
$\Rightarrow a\in\left(-\infty, 4\right]$