Question

The number of solution(s) of the equation $|x-7|-|x+1|=10$ is

Answer 1

$y=|x-7|-|x+1|$
Here the turning points are $-1,7$
$y=\{\begin{array}{cc}-(x-7)+(x+1)& x<-1\\ -(x-7)-(x+1)& -1\le x<7\\ x-7-(x+1)& x\ge 7\end{array}\Rightarrow y=\{\begin{array}{cc}8& x<-1\\ 6-2x& -1\le x<7\\ -8& x\ge 7\end{array}$

When $-1\le x<7$, we get
$10=6-2x\Rightarrow 2x=-4\Rightarrow x=-2$
But $x\in [-1,7)$
Hence, no solutions exist.