Question

The number of integral roots of the equation $|x-1|+|x+2|-|x-3|=4$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (C)

$|x-1|+|x+2|-|x-3|=4$ , has three absolute value expressions, thus we divide the problem into four intervals :
(i) If $x<-2$ then
$-(x-1)-(x+2)+(x-3)-=4\Rightarrow x=-8$
(ii) If $-2\le x<1,$ then
-$(x-1)+1(x+2)+(x-3)=4$
$\Rightarrow x=4\notin [-2,1)$, hence rejected
(iii) If $1\le x<3$, then
$(x-1)+(x+2)+(x-3)=4\Rightarrow x=2$
(iv) If $x\ge 3$, then
$(x-1)+(x+2)-(x-3)=4\Rightarrow x=0\notin [3,\infty )$, hence rejected
$\therefore$ Solution set is $\left\{-8,2\right\}$ and both are integers