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Math Help - Find the solution:

  1. #1
    Member great_math's Avatar
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    Find the solution:

    Find the solution: \frac{(3x-1)^3(x-2)(-5x-2)^2}{(x+2)^7}  > 0
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  2. #2
    Member Henderson's Avatar
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    You have four critical points where things could change: x = \frac{1}{3}, 2, -\frac{2}{5}, -2.

    Plot those four points on a number line and make a sign diagram by plugging in a number from each of the 5 regions you've created to see if you get a positive or a negative.
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  3. #3
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    Hello, great_math!

    Edit: Henderson has already supplied the analysis . . .

    Find the solution: . \frac{(3x-1)^3(x-2)(-5x-2)^2}{(x+2)^7} \;>\; 0

    Consider graphing the function.

    There are x-intercepts at: . x \;=\;\tfrac{1}{3},\;2,\;\text{-}\tfrac{2}{5} (multiplicity 2)

    There is a vertical asymptote: . x \:=\:\text{-}2
    . . Note: there is a horizontal asymptote, y = 0

    Testing values in each interval, we find:

    . . \begin{array}{cc}(\text{-}\infty,\:\text{-}2) & \text{negative} \\ \\[-4mm]<br />
\left(\text{-}2,\:\text{-}\tfrac{2}{5}\right) & \text{positive} \\ \\[-4mm]<br /> <br />
\left(\text{-}\tfrac{2}{5},\:\tfrac{1}{3}\right) & \text{positive} \\ \\[-4mm]<br /> <br />
\left(\tfrac{1}{3},\:2\right) & \text{negative} \\ \\[-4mm]<br />
 (2,\:\infty) & \text{positive} \end{array}

    The graph looks like this:
    Code:
                  :         |
                  :*        |                  *
                  :         |                *   *
                  : *       |*                      *
                  :  *    * |  *            *           *
      - - - - - - : - - o - + - -o- - - - -o- - - - - - - - -
          *       :   -2/5  |   1/3*     * 2
              *   :         |         *
                * :         |
                 *:         |
                 -2         |

    Solution: . \left(\text{-}2,\,\text{-}\tfrac{2}{5}\right) \cup \left(\text{-}\tfrac{2}{5},\,\tfrac{1}{3}\right) \cup (2,\,\infty)

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  4. #4
    o_O
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    As if you've made that with ASCII.
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