If \( \displaystyle\frac{x+3}{2x-1}\lt 1 \), then the solution set contains :
A. \( (-\infty,1/2) \)
B. \( (-\infty,4) \)
C. \( (-4,4) \)
D. \( (4,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and D only
Since the sign of 2x − 1 is unknown, the inequality is not multiplied by it. Bringing everything to one side instead:
\( \displaystyle \frac{x+3}{2x-1}-1\lt 0 \)
\( \displaystyle \Rightarrow\ \frac{x+3-(2x-1)}{2x-1}\lt 0 \)
\( \displaystyle \Rightarrow\ \frac{4-x}{2x-1}\lt 0 \)
\( \displaystyle \frac{x-4}{2x-1}\gt 0 \)
The fraction is positive when numerator and denominator have the same sign; the critical points are \( x=\frac{1}{2} \) and x = 4:
\( \displaystyle x\lt\frac{1}{2}:\ \text{both negative} \)
\( \displaystyle \frac{1}{2}\lt x\lt 4:\ \text{signs differ} \)
\( \displaystyle x\gt 4:\ \text{both positive} \)
So the solution set is \( \left(-\infty,\frac{1}{2}\right)\cup(4,\infty) \).
Checking the statements
(A) \( (-\infty,1/2) \) — Correct. It is one of the two parts of the solution set.
(B) \( (-\infty,4) \) — Incorrect. It includes x = 1, where \( \frac{1+3}{2-1}=4 \), which is not less than 1.
(C) \( (-4,4) \) — Incorrect. It also includes x = 1, where the fraction is 4, not less than 1.
(D) \( (4,\infty) \) — Correct. It is one of the two parts of the solution set.