Question

How many solutions are there to the equation
\[ |x| = -\dfrac 1 2 x + 4?\]

Answer 1

|x| = -1/2 x + 4
First off, you need x < 8, since only then is -1/2 x + 4 >= 0
so now we have
if 0 <= x <= 8 then
x = -1/2 x + 4
3/2 x = 4
x = 8/3
If x < 0 then
-x = -1/2 x + 4
-1/2 x = 4
x = -8
so the solution set is {-8, 8/3}