Let's build a sign chart
color(white)(aaaa)xcolor(white)(aaaaaaa)-oocolor(white)(aaaaaaaaa)1/2color(white)(aaaaaaaaaa)5color(white)(aaaaaaaaa)+oo
color(white)(aaaa)4x-2color(white)(aaaaaaa)-color(white)(aaaaaa)0color(white)(aaaa)+color(white)(aaaaaaaa)+
color(white)(aaaa)-x+5color(white)(aaaaaa)+color(white)(aa)#color(white)(aaaaaaaaa)+#color(white)(aaaaa)0color(white)(aaa)-
color(white)(aaaa)|4x-2|color(white)(aaaaaa)-4x+2color(white)(aa)#color(white)(aaaa)4x-2#color(white)(aaaaa)4x-2
color(white)(aaaa)|-x+5|color(white)(aaaaaaa)x-5color(white)(aa)#color(white)(aaaa)x-5#color(white)(aaaaa)-x+5
Therefore,
In the interval x in (-oo, 1/2)
-4x+2=x-5
5x=7
x=7/5
In the interval x in (1/2, 5)
4x-2=x-5
3x=-3
x=-1
In the interval x in (5, +oo)
4x-2=-x+5
5x=7
x=7/5
Therefore,
the solutions are S={-1, 7/5}
graph{(y-|4x-2|)(y-|-x+5|)=0 [-7.11, 8.69, -0.55, 7.35]}