Question

If S is the set of all real $x$ such that $\frac{2x-1}{{2x}^3+{3x}^2+x}$ is positive, then S contains

• A. $(-\infty ,-\frac{3}{2})$
• B. $(-\frac{3}{2},-\frac{1}{4})$
• C. $(-\frac{1}{4},\frac{1}{2})$
• D. $\left(\frac{1}{2},3\right)$

Answer 1

Answer: (A), (D)

The correct options are
A $(-\infty ,-\frac{3}{2})$
D $\left(\frac{1}{2},3\right)$

$y=\frac{2x-1}{2x^3+3x^2+x}=\frac{2x+1}{x(2x+1)(x+1)}$

$=\frac{{(2x-1)}^2}{(2x+1)(x+1)x(2x-1)}$

For $y$ to be positive $(x+1)(2x+1)x(2x-1)>0$

$\Rightarrow x\in (-\infty ,-1)\cup (\frac{-1}{2},0)\cup (\frac{1}{2},\infty )$

So $xϵ(-\infty ,\frac{-3}{2})\cup (\frac{1}{2},3)$