Given:
\(\frac{2{\text{x}}-3}{3{\text{x}}-7}\)< 0, x ∈ R
Signs of 2x – 3:
2x – 3 = 0 → x = \(\frac{3}{2}\)
(Adding 3 on both the sides and then dividing both sides by 2)
2x – 3 < 0 → x > \(\frac{3}{2}\)
(Adding 3 on both the sides and then dividing both sides by 2)
Signs of 3x – 7:
3x – 7 = 0 → x = \(\frac{7}{3}\)
(Adding 7 on both the sides and then dividing both sides by 3)
3x – 7 < 0 → \(x< \frac{7}{3}\)
(Adding 7 on both the sides and then dividing both sides by 3)
3x – 7 > 0 → \(x> \frac{7}{3}\)
(Adding 7 on both the sides and then dividing both sides by 3)
Zeroes of denominator:
3x – 7 = 0
x = \(\frac{7}{3}\)
(Adding 7 on both the sides and then dividing both sides by 3)
Interval that satisfies the required condition: < 0
\(\frac{3}{2}< x < \frac{7}{3}\)