Question

Solve each of the following system of equations in R.

17. $\frac{2x+1}{7x-1}>5,\frac{x+7}{x-8}>2$

Answer 1

$$\begin{gathered} \frac{2x+1}{7x-1}>5 \\ \Rightarrow \frac{2x+1}{7x-1}-5>0 \\ \Rightarrow \frac{2x+1-35x+5}{7x-1}>0 \\ \Rightarrow \frac{6-33x}{7x-1}>0 \\ \Rightarrow x\in \left(\frac{1}{7},\frac{2}{11}\right)...\left(\mathrm{i}\right) \\ \mathrm{Also},\frac{x+7}{x-8}>2 \\ \Rightarrow \frac{x+7}{x-8}-2>0 \\ \Rightarrow \frac{x+7-2x+16}{x-8}>0 \\ \Rightarrow \frac{23-x}{x-8}>0 \\ \Rightarrow x\in (-\infty ,8)\cup \left(23,\infty \right).....\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequalities}\mathrm{is}\mathrm{the}\mathrm{intersection}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\mathrm{which}\mathrm{is}\mathrm{empty}. \\ \therefore x\in \varnothing \end{gathered}$$