# Solve the equation |3x-4| = |3-4x|Solve the equation |3x-4| = |3-4x|

### 5 Answers | Add Yours

Using the property of absolute value yields:

and

Hence, such that:

Considering yields:

Considering yields:

**Hence, evaluating the solutions to equation yields and .**

As |a|=|-a|, this equation can have many solutions.

We have 4 options that we have to work with.

3x-4=3-4x, 3x-4=4x-3, 4-3x=3-4x and 4-3x=4x-3

For each of the relations the value of x that we get is:

3x-4=3-4x => 7x=7 or x=1

3x-4=4x-3 => x=-1

4-3x=3-4x => x=-1

4-3x=4x-3 => 7x=7 or x=1

Therefore the solutions for **x are 1 and -1**

squaring on both side u will get

9x2 -12x + 16 = 9 - 12x + 16x2

simplifying will result to

7x2 = 7

x2 = 1

sq root on both sides will give u

x= +1, -1

To solve |3x-4| = |3-4x|

If 3x > 4, or x > 4/3

Then 3x-4 = |3-4x| = 4x-4 , as

3x-4 =4x-3 Or

3-4 = 4x-3x =x . So x=-1 which is inconsistent.

If x < 4/3 and x >3/4

LHS = 4-3x and RHS = |3-4x| = 4x-3

4+3 = 4x+3x

7 = 7x Or x =1.

When x <3/4,

LHS : 4-3x and RHS = 3-4x

4-3x = 3-4x

4x-3x = 3-4 =

x =-1 which is consistent with x <3/4.

Therefore x =-1 or x= 1.

To solve the equation, we'll consider 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}.**