Solution set of the linear inequation \( \displaystyle\frac{3}{x-2}\lt 1 \) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( (-\infty,2)\cup(5,\infty) \)
Since the sign of x − 2 is not known, the inequation cannot be multiplied by x − 2. Bringing everything to one side:
\( \displaystyle \frac{3}{x-2}-1\lt 0 \)
\( \displaystyle \frac{5-x}{x-2}\lt 0 \)
\( \displaystyle \frac{x-5}{x-2}\gt 0 \)
The fraction is positive when the numerator and denominator have the same sign, that is, when x < 2 or x > 5. At x = 5 the left side equals 1, which is not less than 1, so 5 is excluded.
Why the other options are incorrect
Option (1) \( (2,5] \) — Incorrect. For 2 < x < 5, \( \frac{3}{x-2}\gt 1 \); and at x = 5 it equals 1.
Option (2) \( (2,5) \) — Incorrect. This comes from multiplying by x − 2 as if it were positive; it is where \( \frac{3}{x-2}\gt 1 \).
Option (3) \( (-\infty,2)\cup[5,\infty) \) — Incorrect. At x = 5, \( \frac{3}{x-2}=1 \), which is not less than 1.