Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and


Write the set of values of $x$ satisfying the inequations $5 x+2<3 x+8$ and $\frac{x+2}{x-1}<4$.


We have:

$5 x+2<3 x+8$ and $\frac{x+2}{x-1}<4$

$\Rightarrow 2 x<6$ and $\frac{x+2}{x-1}-4<0$

$\Rightarrow x<3$ and $\frac{x+2-4 x+4}{x-1}<0$

$\Rightarrow x \in(-\infty, 3)$ and $\frac{-3 x+6}{x-1}<0$

$\Rightarrow x \in(-\infty, 3)$ and $\frac{-x+2}{x-1}<0$

For $\frac{-x+2}{x-1}<0$, critical points are 1 and $2 .$

$\Rightarrow x \in(2, \infty) \cup(-\infty, 1)$

$\therefore x \in(-\infty, 1) \cup(2,3)$

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