1. ## |2x+7|-|6-3x|=8

How do I solve this equation involving absolute values?
I tried to solve it with these two equations and their consequent pairs (equations with the negative on the right side and the inequality flipped) |2x+7| 8 and 8 |6-3x| 8
but they don't seem to be working.

If you can help, please comment.

2. ## Re: |2x+7|-|6-3x|=8

Originally Posted by firefly2552025
How do I solve this equation involving absolute values?
I tried to solve it with these two equations and their consequent pairs (equations with the negative on the right side and the inequality flipped) |2x+7| 8 and 8 |6-3x| 8
but they don't seem to be working.

If you can help, please comment.
Start by looking at critical points: in this case values of x such that 2x + 7 = 0 and 6 - 3x = 0. Then you set up equation to test where the expression can be solved. For example, on -7/2 < x < 2 we would have the equation -(2x + 7) - (6 - 3x) = 8. What is x for this? That and other similar equations will give you the values of x for your equation.

-Dan

Made an oopsie! It should read (2x + 7) - (6 - 3x) = 8. Thanks to JeffM for the catch.

3. ## Re: |2x+7|-|6-3x|=8

Originally Posted by firefly2552025
How do I solve this equation involving absolute values?
I tried to solve it with these two equations and their consequent pairs (equations with the negative on the right side and the inequality flipped) |2x+7| 8 and 8 |6-3x| 8
but they don't seem to be working.

If you can help, please comment.
Start by identifying the values of x where the expressions inside the absolute value operator are non-negative.

$6 - 3x \ge 0 \implies x \le 2 \implies |6 - 3x| = 6 - 3x.\ And\ x > 2 \implies |6 - 3x| = 3x - 6.$

$2x + 7 \ge 0 \implies x \ge -3.5 \implies |2x + 7| = 2x + 7.\ And\ x < -3.5 \implies |2x + 7| = - 2x - 7.$

That gives three different cases to examine: $x < -3.5,\ -3.5 \le x < 2,\ and\ x \ge 2.$

$|2x - 7| - |6 - 3x| = 8\ and\ x < - 3.5 = 8 \implies (-2x - 7) - (6 - 3x) = 8 \implies x -13 = 8 \implies x = 21 \not < -3.5.$ Not a solution.

Now you look at the other cases.

4. ## Re: |2x+7|-|6-3x|=8

Originally Posted by JeffM
$|2x - 7| - |6 - 3x| = 8\ and\ x < - 3.5 = 8 \implies (-2x - 7) - (6 - 3x) = 8 \implies x -13 = 8 \implies x = 21 \not < -3.5.$ Not a solution.
That should be: $|2x + 7| - |6 - 3x| = 8\ and\ x < - 3.5 \implies (-2x - 7) - (6 - 3x) = 8 \implies x -13 = 8 \implies x = 21 \not < -3.5.$ Not a solution.

5. ## Re: |2x+7|-|6-3x|=8

I solved it with help of determining which set it could be in. Thanks.