Solution of the inequality \(\displaystyle\frac{2x+3}{4x-5}\ge 0\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\left(-\infty,-\frac{3}{2}\right]\left(\frac{5}{4},\infty\right)\)
The numerator is zero at \(x = -\frac{3}{2}\) and the denominator is zero at \(x = \frac{5}{4}\). These critical points split the line into three intervals; the sign of the fraction in each is
\( \displaystyle x \lt -\frac{3}{2}: \text{ numerator } -, \text{ denominator } -, \text{ fraction } + \)
\( \displaystyle -\frac{3}{2} \lt x \lt \frac{5}{4}: \text{ numerator } +, \text{ denominator } -, \text{ fraction } - \)
\( \displaystyle x \gt \frac{5}{4}: \text{ numerator } +, \text{ denominator } +, \text{ fraction } + \)
At \(x = -\frac{3}{2}\) the fraction is 0, which satisfies \(\ge 0\), so this point is included. At \(x = \frac{5}{4}\) the fraction is not defined, so this point is excluded. Hence
\( \displaystyle x \in \left(-\infty, -\frac{3}{2}\right] \cup \left(\frac{5}{4}, \infty\right) \)
The options write the two intervals side by side, meaning their union; this is Option (4).
Why the other options are incorrect
Option (1) \(\displaystyle\left(-\frac{3}{2},\frac{5}{4}\right)\) — Incorrect. Here the fraction is negative; at \(x = 0\) it is \(-\frac{3}{5}\).
Option (2) \(\displaystyle\left(-\infty,-\frac{3}{2}\right]\left[\frac{5}{4},\infty\right)\) — Incorrect. It includes \(x = \frac{5}{4}\), where the denominator is zero.
Option (3) \(\displaystyle\left[-\frac{3}{2},\frac{5}{4}\right)\) — Incorrect. This is where the fraction is \(\le 0\), not \(\ge 0\).