# Sove for x in |2x-6|=|4-5x| .

hala718 | Certified Educator

l 2x - 6l = l 4-5x l

That means we have 4 possible ways to solve:

1)  (2x - 6) = (4 - 5x)

==> 7x = 10

==> x = 10/7

2) -(2x-6) = (4-5x)

==> -2x + 6 = 4 - 5x

==> 3x = -2

==> x = -2/3

3) (2x-6) = - (4- 5x)

==> 2x - 6 = -4 + 5x

==> -3x = 2

==> x = -2/3

4) -(2x-6) = -(4-5x)

==> -2x + 6 = -4 + 5x

==> - 7x = -10

==> x = 10/7

==> x = { 10/7, -2/3}

giorgiana1976 | Student

To solve the equality, we'll consider 4 cases of study.

The expressions 2x-6 and 4-5x can be either positive or negative.

Case 1:

2x-6 = 4-5x

Case 2:

-(2x-6) = 4-5x

Case 3:

2x-6 = -(4-5x)

Case 4:

-(2x-6) = -(4-5x)

We notive that 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:

2x-6 = 4-5x

We'll isolate x to the left side:

2x + 5x = 4+6

7x = 10

We'll divide by 7:

x = 10/7

Case 2:

-(2x-6) = 4-5x

We'll remove the brackets:

-2x + 6 = 4 - 5x

We'll isolate x to the left side:

-2x + 5x = 4-6

3x = -2

We'll divide by 3:

x = -2/3

The solutions of the equation are: {-2/3 ; 10/7}.

neela | Student

When 2x-6 > 6, Or x > 3,the equation becomes |2x-6| = 2x-6.

|4-5x| = 5x-4 as 4-5x is -ve when x >3.

So the equation is 2x-6 = 5x-4

-6+4 = 5x-2x

-2 = 3x

x = -2/3.

When 2x-6 <6, Or x < 3, |2x-6| = 6-2x and |4-5x| = 5x-4 , till x is >4/5.

6-2x = 5x -4  for  4/5 <x < 3

6+4 =5x+2x = 7x, x = 10/7  which is valid for 4/5 < x <6.

When x < 4/5:

6-2x  = 4-5x

3x = 4-2  Or x = -2/3 is valid  for x <4/5.

So x =-2/3 or x = 10/7