Question

The set of values of $x$ which satisfy the inequations $5x+2<3x+8$ and $\frac{x+2}{x-1}<4$ is

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

Answer 1

Answer: (D)

We have, $5x+2<3x+8$ and $\frac{x+2}{x-1}<4$
$\Rightarrow x<3$ and $\frac{\left(x+2\right)\left(x-1\right)}{{\left(x-1\right)}^2}<4$, $x\ne 1$
$\Rightarrow x<3$ and $\left(x+2\right)\left(x-1\right)<4x^2-8x+4$, $x\ne 1$
$\Rightarrow x<3$ and $3x^2-9x+6>0$, $x\ne 1$
$\Rightarrow x<3$ and $x^2-3x+2>0$, $x\ne 1$
$\Rightarrow x<3$ and $\left(x-1\right)\left(x-2\right)>0$, $x\ne 1$
$\Rightarrow x<3$ and $(x<1$ or $x>2)$
$\Rightarrow x\in \left(-\infty ,1\right)\cup \left(2,3\right)$.