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Math Help - |2x+7|-|6-3x|=8

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

    Quote Originally Posted by firefly2552025 View Post
    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.
    Last edited by topsquark; June 29th 2014 at 10:55 AM. Reason: Made an oopsie!
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  3. #3
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    Re: |2x+7|-|6-3x|=8

    Quote Originally Posted by firefly2552025 View Post
    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.
    Last edited by JeffM; June 28th 2014 at 10:13 PM.
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    Re: |2x+7|-|6-3x|=8

    Quote Originally Posted by JeffM View Post
    $|2x - 7| - |6 - 3x| = 8\ and\ x < - 3.5 = 8 \implies (-2x - 7) - (6 - 3x) = 8 \implies x -13 = 8 \implies x = 21 \not < -3.5.$ 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.
    Last edited by JeffM; June 29th 2014 at 08:08 AM.
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    Re: |2x+7|-|6-3x|=8

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