The solution set of the inequality \(\displaystyle\frac{2x+3}{x-1} \lt 0\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x \in \left(-\displaystyle\frac{3}{2}, 1\right)\)
A fraction is negative when the numerator and the denominator have opposite signs. The critical points are where each is zero:
\( \displaystyle 2x + 3 = 0 \;\Rightarrow\; x = -\frac{3}{2} \)
\(x - 1 = 0 \;\Rightarrow\; x = 1\)
Checking the sign of the fraction in each interval:
\( \displaystyle x \lt -\frac{3}{2}: \ \text{both negative, fraction positive} \)
\( \displaystyle -\frac{3}{2} \lt x \lt 1: \ \text{numerator positive, denominator negative, fraction negative} \)
\(x \gt 1: \ \text{both positive, fraction positive}\)
At \(x = -\frac{3}{2}\) the fraction is 0, and at x = 1 it is undefined, so both ends are excluded. The solution set is \(\left(-\frac{3}{2}, 1\right)\).