for what values of x does each hold?

Question

/x-2/ = x-2

/2x-3/ = 3-2x

Answer 1

|x - 2| = x - 2 if x - 2 >= 0, so x >= 2

|2x - 3| = 3 - 2x if 2x - 3 <= 0, xo x <= 3/2

In general, |y| = y if y >= 0 and |y| = -y if y <= 0. Note that 3 - 2x = -(2x - 3).

Answer 2

1. x-2=x-2. This works all the time, so there are infinitely many answers when looking at the positive case.
x-2=-(x-2) would show that 2x=4 so x=2.

2. 2x-3 = 3-2x. 4x=6, so x=1.5 or 3/2
2x-3 = -(3-2x). 2x-3 = -3+2x. This works all the time so there are infinitely many answers.

Answer 3

PROBLEM 1:

|x-2| = x-2 implies

x-2 = x-2 Eq'n 1a
-(x-2) = x-2 Eq'n 1b

Eq'n 1a gives us x = 0 while
Eq'n 1b gives us 2x - 4 = 0 or x = 2

PROBLEM 2:

|2x-3| = 3-2x implies

2x - 3 = 3-2x Eq'n 2a
-(2x-3) = 3-2x Eq'n 2b

Eq'n 2a gives us 4x = 6 or x = 3/2 = 1 ½ while
Eq’n 2b gives us x = 0