Question

If $|x-1|+|x-3|\le 8$, then the values of $x$ lie in the interval

• A. $(-\infty ,-2)$
• B. $[-2,6]$
• C. $(-3,7)$
• D. $(-2,\infty )$
• E. $[6,\infty )$

Answer 1

The correct option is B $[-2,6]$

Given inequality is
$|x-1|+|x-3|\le 8$

Case I $x<1$
$|x-1|+|x-3|\le 8$
$\Rightarrow -(x-1)-(x-3)\le 8$
$\Rightarrow -2x+4\le 8$
$\Rightarrow -2x\le 4\Rightarrow x\ge -2$

Interval $[-2,1)$

Case II $1\le x<3$
$\Rightarrow x-1-x+3\le 8\Rightarrow 2\le 8$

Interval $[1,3]$

Case III $x\ge 3$
$\Rightarrow x-1+x-3\le 8\Rightarrow x\le 6$

Interval $[3,6]$

From all the interval
values of $x$ lie in the interval $[-2,6]$
Hence (b) is the correct option