Question

Let $A=\{x:xisrealsatisfying\frac{x}{4}<\frac{5x-2}{3}-\frac{(7x-3)}{5}\}$ and
$B=\{x:xisrealsatisfying\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}\}$, then

• A. $A\cap B=\varphi$
• B. $A\cup B=Z$,Z is an integer
• C. $A\cap B=A$
• D. $A\cap B=B$

Answer 1

The correct option is A $A\cap B=\varphi$

$\frac{x}{4}<\frac{5x-2}{3}-\frac{(7x-3)}{5}\Rightarrow x>4\Rightarrow xϵ(4,\infty )\therefore A=(4,\infty )\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}\Rightarrow \frac{-(3x-2)}{(2x+1)(x+2)(x+1)}>0\Rightarrow \frac{3x+2}{(2x+1)(x+2)(x+1)}<0\Rightarrow xϵ(-2,-1)\cup (\frac{-2}{3},\frac{-1}{2})\therefore B=(-2,-1)\cup (\frac{-2}{3},\frac{-1}{2})\Rightarrow A\cap B=\varphi$