how to solve modulus (2x - 1/x - 1)>2?

if the given inequality is: $| \frac{2 x - 1}{x - 1} | > 2$ .............(1)

if $| x | > a \Rightarrow x > a o r x < - a$

therefore for given in equality two cases will arise:

case I :

$\frac{2 x - 1}{x - 1} > 2 \frac{2 x - 1}{x - 1} - 2 > 0 \frac{2 x - 1 - 2 (x - 1)}{x - 1} > 0 \frac{2 x - 1 - 2 x + 2}{x - 1} > 0 \frac{1}{x - 1} > 0 \Rightarrow x > 1$ ............(2)

case II:

$\frac{2 x - 1}{x - 1} < - 2 \frac{2 x - 1}{x - 1} + 2 < 0 \frac{2 x - 1 + 2 (x - 1)}{x - 1} < 0 \frac{2 x - 1 + 2 x - 2}{x - 1} < 0 \frac{4 x - 3}{x - 1} < 0 \Rightarrow \frac{4 (x - \frac{3}{4})}{x - 1} < 0$

first plot the number 3/4 and 1 on the number line. and find solution by wavy curve method.

thus $x \in (\frac{3}{4}, 1)$ ............(3)

so from (2) and (3), $x \in (\frac{3}{4}, \infty) - {1}$

hope this helps you.

cheers!!

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