Question

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

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ 5x+2<3x+8\mathrm{and}\frac{x+2}{x-1}<4 \\ \Rightarrow 2x<6\mathrm{and}\frac{x+2}{x-1}-4<0 \\ \Rightarrow x<3\mathrm{and}\frac{x+2-4x+4}{x-1}<0 \\ \Rightarrow x\in (-\infty ,3)\mathrm{and}\frac{-3x+6}{x-1}<0 \\ \Rightarrow x\in (-\infty ,3)\mathrm{and}\frac{-x+2}{x-1}<0 \\ \mathrm{For}\frac{-x+2}{x-1}<0,\mathrm{critical}\mathrm{points}\mathrm{are}1\mathrm{and}2. \\ \Rightarrow x\in (2,\infty )\cup (-\infty ,1) \\ \therefore x\in (-\infty ,1)\cup (2,3) \end{gathered}$$