Question

Complete values of $a$ for which the equation $|x-1|+|x-2|+x-a>0$ has two solutions, is

• A. $a\in (-\infty ,-2)$
• B. $a\in (2,\infty )$
• C. $a\in (-5,\infty )$
• D. $a\in (-2,\infty )$

Answer 1

The correct option is B $a\in (2,\infty )$

$a=|x-1|+|x-2|+x$
$\Rightarrow a=\{\begin{array}{ccc}-(x-1)-(x-2)+x=-x+3,& x<1& \\ (x-1)-(x-2)+x=x+1,& 1\le x<2& \\ (x-1)+(x-2)+x=3x-3& x\ge 2& \end{array}$
So $a$ must be greater than 2 for two solutions.