Question

Solve each of the following system of equations in R.

11. 4x − 1 ≤ 0, 3 − 4x < 0

Answer 1

$$\begin{gathered} \mathrm{We}have,4x-1\le 0 \\ \Rightarrow 4x\le 1 \\ \Rightarrow x\le \frac{1}{4}(\mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{by}4) \\ \Rightarrow x\in (-\infty ,\frac{1}{4}]...\left(\mathrm{i}\right) \\ \mathrm{Also},3-4x<0 \\ \Rightarrow 0>3-4x \\ \Rightarrow 4x>3 \\ \Rightarrow x>\frac{3}{4}\mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}4 \\ \Rightarrow x\in \left(\frac{3}{4},\infty \right)...\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequalities}\mathrm{is}\mathrm{the}\mathrm{intersection}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right). \\ \mathrm{But},\left(-\infty \frac{1}{4}\right)\cap \left(\frac{3}{4},\infty \right)=\mathrm{\varphi } \\ \mathrm{Thus},\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequations}\mathrm{has}\mathrm{no}\mathrm{solution}. \end{gathered}$$