For |2x + 3| - |2x -5| = 6 , what is x? How do I solve the absolute functions?

Answer 1

|2x+3| - |2x-5| = 6

|2x+3| = 6 + |2x-5| <--- adds |2x+5| to both sides

|2x+3| - 6 = |2x-5| <--- subtracts 6 from both sides

|2x-5| = |2x+3| - 6 <--- symmetric property of equality

2x-5 = |2x+3|-6 OR 2x-5 = - ( |2x+3|-6) <---- rule from absolute value: |z|=k means z=k or z=-k

First equation:

2x-5 = |2x+3|-6

2x+1 = |2x+3|

|2x+3| = 2x+1

2x+3 = 2x+1 or 2x+3 = -(2x+1)

1st eq: 3 = 1 is a dead end

2nd eq: 2x + 3 = -2x - 1

4x + 3 = -1

4x = -4

x = -1

does not work; FAILS

second equation:

2x-5 = - ( |2x+3|-6)

2x-5 = -|2x+3| + 6

2x - 11 = |2x+3|

|2x+3| = 2x - 11

1st eq: 2x+3 = 2x - 11

3 = -11 is a dead end

2nd eq:

2x +3 = -2x + 11

4x + 3 = 11

4x = 8

x = 2

x=2 is the only solution that works

Answer 2

:Four possibilities exist:

1) both arguments are positive

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

2) both arguments are negative

-(2x - 3) - [-(2x - 5)] = 6

3) first arguments is positive, second negative

2x - 3 - [-(2x - 5)] = 6

4) first argument is negative, second positive

-(2x - 3) - (2x - 5) = 8