You may also use the absolute value definition, such that:

`|3 - 4x| = {(3 - 4x, 3 - 4x >= 0),(4x - 3,3 - 4x < 0):}`

`|3 - 4x| = {(3 - 4x, x <= 3/4),(4x - 3, x > 3/4):}`

`|3x - 4| = {(3x - 4, x >= 4/3),(4 - 3x, x < 4/3):}`

`|3 - 4x| - |3x - 4| = 0`

`3 - 4x - 4 + 3x = 0 => -x = 4 - 3 => x = -1`

`4x - 3 - 4 + 3x = 0 => 7x = 7 => x = 1`

`4x - 3 - 3x + 4 = 0 => x = -1 < 4/3`

**Hence, evaluating the solutions to the given absolute value equation yields `x = -1 , x = 1.` **

The equation |3-4x|-|3x-4|=0 has to be solved.

The absolute value of an expression x, |x| is equal to x when x >= 0 and it is equal to -x when x < 0.

Here, there are 2 absolute values in the equation. This gives the following equations that have to be solved:

3 - 4x + 3x - 4 = 0

-x = 1

x = -1

3-4x - 3x + 4 = 0

-7x = -7

x = 1

The solution of the equation |3-4x|-|3x-4|=0 is x = 1 and x = -1

We'll solve the equation considering 4 cases of study.

The expressions 3x-4 and 3-4x can be either positive or negative.

Case 1:

3x-4 = 3-4x

Case 2:

-(3x-4) = 3-4x

Case 3:

3x-4 = -(3-4x)

Case 4:

-(3x-4) = -(3-4x)

The Case 2 and Case 3 will have the same solution. Also Case 1 and Case 4 will have the same solution. So, we'll solve just Case 1 and Case 2:

Case 1:

3x-4 = 3-4x

We'll isolate x to the left side:

3x + 4x = 3+4

7x = 7

We'll divide by 7:

**x = 1**

Case 2:

-(3x-4) = 3-4x

We'll remove the brackets:

-3x +4 = 3-4x

We'll isolate x to the left side:

-3x + 4x = 3-4

x = -1

The solutions of the equation are: {-1 ; 1}.