$Solve for x : (x - 2)^{3} (x - 3) (x - 5)^{2} < 0$

(

- \infty, 2) \cup (3, \infty)

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(2, 3)

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$(3, \infty)$

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$(5, \infty)$

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Solution

The correct option is B (2, 3)
Mark the zeroes of expression $(x - 2)^{3} (x - 3) (x - 5)^{2}$ and apply the wavy-curve method

Zeroes are $\Rightarrow$ x = 2,3,5

Explanation

$\begin{matrix} I n t e r v a l & S i g n o f & O v e r a l l s i g n o f p r o d u c t (x - 2)^{3} & (x - 3) & (x - 5)^{2} x > 5 & + & + & + & + 3 < x < 5 & + & + & + & + 2 < x < 3 & + & - & + & - x < 2 & - & - & + & + \end{matrix}$

Thus,by above table, we arrived at the number line sign-scheme.

So, the value of x for which

$(x - 2)^{3}$ $(x - 3) (x - 5)^{2}$ < 0

x ∈ (2,3)

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