Question

Which of the following are solutions to | x-1|= 5x + 2 ? Check all that apply.

Answer 1

By definition of absolute value, if $x-1\ge0$, then $|x-1|=x-1$, so

$|x-1|=5x+2\implies x-1=5x+2\implies4x=-3\implies x=-\dfrac34$

On the other hand, if $x-1$, then $|x-1|=-(x-1)=1-x$, and

$|x-1|=5x+2\implies1-x=5x+2\implies6x=-1\implies x=-\dfrac16$