Given,
|(2x-1)/(x-1)| > 2
x ≠ 1,
As it will lead equation unmeaningful.
Now,
On subtracting 2 from both the sides, we get–
\(|\frac{2x-1}{x-1}|\) - 2 > 0
Now,
3 case arises :
Case 1 : 1 < x < ∞
For this case,
|2x–1| = 2x–1 and |x–1| = x–1
⇒ x ∈ (1, ∞) …(1)
Case 2 : \(\frac{1}{2}\) < x < 1
For this case:
|2x–1| = 2x–1 and |x–1| = –(x–1)
⇒ x ∈ (\(\frac{3}{4}\), 1) …(2)
Case 3 : -∞ < x < \(\frac{1}{2}\)
For this case :
|2x–1| = –(2x–1) and
|x–1| = –(x–1)
Which is not possible,
Hence,
This will give no solution.
⇒ x ∈ (\(\frac{3}{4}\), 1) ∪ (1, ∞) (from 1 and 2)
We can verify the answers using graph as well.
