Question

Solve each of the following system of equations in R.

12. x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Answer 1

$$\begin{gathered} x+5>2\left(x+1\right) \\ \Rightarrow x+5>2x+2 \\ \Rightarrow 2x+2<x+5 \\ \Rightarrow 2x-x<5-2 \\ \Rightarrow x<3 \\ \Rightarrow x\in (-\infty ,3)...\left(\mathrm{i}\right) \\ \mathrm{Also},2-x<3\left(x+2\right) \\ \Rightarrow 2-x<3x+6 \\ \Rightarrow 3x+6>2-x \\ \Rightarrow 3x+x>2-6 \\ \Rightarrow 4x>-4 \\ \Rightarrow x>-1 \\ \Rightarrow x\in (-1,\infty )...\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequations}\mathrm{is}\mathrm{the}\mathrm{intersection}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right). \\ (-\infty ,3)\cap (-1,\infty )=(-1,3) \\ \mathrm{Thus},\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequations}\mathrm{is}(-1,3). \end{gathered}$$