Question

If $S$ is the set of all real $x$ and 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. $(\frac{1}{2},3)$

Answer 1

Answer: (A), (D)

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

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

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

For $y$ to be positive,

$(2x+1)(x+1)x(2x-1)>0$

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