Let f (x) = x + |2x - 1|. find all solutions to the equation f (x) = 8. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.)

Hi!

f (x) = x + |2x - 1|

We have the function at hand, so if the function is equal to 8, then x + |2x - 1| is equal to 8 too:

x + |2x - 1| = 8

Now we can express the module so:

|2x - 1| = 8-x

It means we'll have 2 options:

I : 2x-1 = 8-x

II : 2x-2 = - (8-x)

I : 2x-1 = 8-x | + x

2x-1+x = 8-x+x

3x-1 = 8 | + 1

3x-1+1 = 8+1

3x = 9 | : 3

x = 3

II : 2x-1 = - (8-x)

2x-1 = - 8+x | - x

2x-x-1 = - 8+x-x

x-1 = - 8 | + 1

x-1+1 = - 8+1

x = - 7

Solutions : 3,-7