Question

The complete set of values of $x$ satisfying $5x+2<3x+8$ and $\frac{x+2}{x-1}<4,$ is

• A. $(-\infty ,2)\cup (3,\infty )$
• B. $(2,3)$
• C. $(-\infty ,3)$
• D. $(-\infty ,1)\cup (2,3)$

Answer 1

The correct option is D $(-\infty ,1)\cup (2,3)$

Given :
$5x+2<3x+8\Rightarrow 2x<6\Rightarrow x<3\cdots \left(1\right)$

Also, $\frac{x+2}{x-1}<4$
$\Rightarrow \frac{x+2}{x-1}-4<0\Rightarrow \frac{-3x+6}{x-1}<0\Rightarrow \frac{x-2}{x-1}>0\Rightarrow x\in (-\infty ,1)\cup (2,\infty )\cdots \left(2\right)$

Fron equations $\left(1\right)$ and $\left(2\right),$ we get
$x\in (-\infty ,1)\cup (2,3)$