Question

Solve the following system of equations in R.
$\left|\frac{2x-1}{x-1}\right|>2$

Answer 1

We have,

$\frac{|2x-1|}{x-1}-2>0\frac{|2x-1|-2(x-1)}{x-1}>0\frac{|2x-1|-2x+2}{x-1}>0...\left(i\right)CaseI:when|2x-1|\ge 0i.e.,2x-1\ge 02x\ge 1x\ge \frac{1}{2}\Rightarrow |2x-1|-2x+2>0andx-1>0\Rightarrow 2x-1-2x+2>0andx>1\Rightarrow x>1...\left(ii\right)CaseII:when|2x-1|<0i.e.,2x-1<02x<1x<\frac{1}{2}\Rightarrow -(2x-1)-2x+2>0andx<1\Rightarrow -4+3>0\Rightarrow -x>-\frac{3}{4}\Rightarrow x<\frac{3}{4}andx<1xϵ(\frac{3}{4},1)...\left(iii\right)\text{Combining (ii) and (iii) we get}(\frac{3}{4},1)\cup (1,\infty )\text{as the solution set.}$