Hello, great_math!
Edit: Henderson has already supplied the analysis . . .
Find the solution: .$\displaystyle \frac{(3x-1)^3(x-2)(-5x-2)^2}{(x+2)^7} \;>\; 0$
Consider graphing the function.
There are x-intercepts at: .$\displaystyle x \;=\;\tfrac{1}{3},\;2,\;\text{-}\tfrac{2}{5}$ (multiplicity 2)
There is a vertical asymptote: .$\displaystyle x \:=\:\text{-}2$
. . Note: there is a horizontal asymptote, $\displaystyle y = 0$
Testing values in each interval, we find:
. . $\displaystyle \begin{array}{cc}(\text{-}\infty,\:\text{-}2) & \text{negative} \\ \\[-4mm]
\left(\text{-}2,\:\text{-}\tfrac{2}{5}\right) & \text{positive} \\ \\[-4mm]
\left(\text{-}\tfrac{2}{5},\:\tfrac{1}{3}\right) & \text{positive} \\ \\[-4mm]
\left(\tfrac{1}{3},\:2\right) & \text{negative} \\ \\[-4mm]
(2,\:\infty) & \text{positive} \end{array}$
The graph looks like this: Code:
: |
:* | *
: | * *
: * |* *
: * * | * * *
- - - - - - : - - o - + - -o- - - - -o- - - - - - - - -
* : -2/5 | 1/3* * 2
* : | *
* : |
*: |
-2 |
Solution: .$\displaystyle \left(\text{-}2,\,\text{-}\tfrac{2}{5}\right) \cup \left(\text{-}\tfrac{2}{5},\,\tfrac{1}{3}\right) \cup (2,\,\infty) $