Given,
1/(x-1) ≤ 2
For this inequation to be true,
There are two possible cases.
i. 2x – 3 ≥ 0 and x – 1 > 0
⇒ 2x – 3 + 3 ≥ 0 + 3 and
x – 1 + 1 > 0 + 1
⇒ 2x ≥ 3 and x > 1
⇒ x ≥ \(\frac{3}{2}\) and x > 1
ii. 2x – 3 ≤ 0 and x – 1 < 0
⇒ 2x – 3 + 3 ≤ 0 + 3 and
x – 1 + 1 < 0 + 1
⇒ 2x ≤ 3 and x < 1
Hence,
x ∈ (–∞, 1)
Thus,
The solution of the given inequation is ( -∞,1) ∪ [\(\frac{3}{2}\),∞).
