Question

Solve each of the following system of equations in R.

14. $5x-7<3\left(x+3\right),1-\frac{3x}{2}\ge x-4$

Answer 1

$$\begin{gathered} 5x-7<3\left(x+3\right) \\ \Rightarrow 5x-7<3x+9 \\ \Rightarrow 5x-3x<9+7 \\ \Rightarrow 2x<16 \\ \Rightarrow x<8 \\ \Rightarrow x\in \left(-\infty ,8\right)...\left(\mathrm{i}\right) \\ \mathrm{Also},1-\frac{3x}{2}\ge x-4 \\ \Rightarrow x-4\le 1-\frac{3x}{2} \\ \Rightarrow x+\frac{3x}{2}\le 1+4 \\ \Rightarrow \frac{2x+3x}{2}\le 5 \\ \Rightarrow 5x\le 10 \\ \Rightarrow x\le 2 \\ \Rightarrow x\in (-\infty ,2]...\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequalities}\mathrm{is}\text{the}\mathrm{intersection}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right). \\ \left(-\infty ,8\right)\cap (-\infty ,2]=(-\infty ,2] \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequalities}\mathrm{is}(-\infty ,2]. \end{gathered}$$