Question

Solve each of the following system of equations in R.

25. $\frac{1}{\left|x\right|-3}<\frac{1}{2}$

Answer 1

$$\begin{gathered} \mathrm{We}have,\frac{1}{\left|x\right|-3}<\frac{1}{2} \\ \Rightarrow \frac{1}{\left|x\right|-3}-\frac{1}{2}<0 \\ \mathrm{Now},\mathrm{two}\mathrm{cases}\mathrm{arises}. \\ \mathrm{CASE}1:x>0 \\ \mathrm{Then},\left|x\right|=x \\ \therefore \frac{1}{\left|x\right|-3}-\frac{1}{2}<0 \\ \Rightarrow \frac{1}{x-3}-\frac{1}{2}<0 \\ \Rightarrow \frac{2-x+3}{2x-6}<0 \\ \Rightarrow \frac{-x+5}{2x-6}<0 \\ \Rightarrow x\in (-\infty ,3)\cup (5,\infty ) \\ \mathrm{But}\mathrm{in}\mathrm{this}\mathrm{case}x\mathrm{is}\mathrm{positive} \\ \therefore x\in (0,3)\cup (5,\infty )....\left(\mathrm{i}\right) \\ \mathrm{CASE}2:\mathrm{When}x<0 \\ \left|x\right|=-x \\ \therefore \frac{1}{\left|x\right|-3}-\frac{1}{2}<0 \\ \Rightarrow \frac{1}{-x-3}-\frac{1}{2}<0 \\ \Rightarrow \frac{2+x+3}{-2x-6}<0 \\ \Rightarrow \frac{x+5}{-2x-6}<0 \\ \Rightarrow x\in (-\infty ,-5)\cup (-3,\infty ) \\ \mathrm{But}\mathrm{in}\mathrm{this}\mathrm{case}x\mathrm{is}\mathrm{negative} \\ \therefore x\in (-\infty ,-5)\cup (-3,0)....\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{inequation}\mathrm{is}\mathrm{the}\mathrm{union}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right). \\ \therefore x\in (-\infty ,-5)\cup (-3,3)\cup (5,\infty ) \end{gathered}$$