Q. Draw the sign scheme and find the solution
$\frac{{\left(x+1\right)}^3{\left(x-2\right)}^2}{\left(x-5\right)}\ge 0$

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \frac{{\left(\mathrm{x}+1\right)}^3{\left(\mathrm{x}-2\right)}^2}{\left(\mathrm{x}-5\right)}\ge 0 \\ \mathrm{As} \left(\mathrm{x}-2\right) \mathrm{has} \mathrm{even} \mathrm{power} , \mathrm{hence} \mathrm{there} \mathrm{will} \mathrm{no} \mathrm{sign} \mathrm{change} \mathrm{about} \mathrm{x}=2. \\ \mathrm{Now} \mathrm{we} \mathrm{use} \mathrm{method} \mathrm{of} \mathrm{interval} \\ \left(-\infty \right)+++++\left(-1\right)-----\left(2\right)----\left(5\right)++++\left(\infty \right) \\ \mathrm{We} \mathrm{need} \ge 0 \\ \mathrm{Hence} \\ \mathrm{x}\in (-\infty ,-1]\cup [2,5) \end{gathered}$$

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