Question

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

• A. (-∞, 2)∪(3, ∞)
• B. (2, 3)
• C. (3, ∞)
• D. (5, ∞)

Answer 1

The correct option is B

(2, 3)

Mark the zeroes of expression $(x-2{)}^3(x-3\left)\right(x-5{)}^2$ and apply the wavy-curve method

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

Explanation

$$\begin{array}{ccccc}Interval& & Signof& & Overallsignofproduct\\ & (x-2{)}^3& (x-3)& (x-5{)}^2& \\ x>5& +& +& +& +\\ 3<x<5& +& +& +& +\\ 2<x<3& +& -& +& -\\ x<2& -& -& +& +\end{array}$$

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)