The number of integral solutions \(x\) of \(\log_{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2x-3}\right)^{2} \geq 0\) is

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Basic Mathematics

The number of integral solutions

x

lo g_{(x + \frac{7}{2})} (\frac{x - 7}{2 x - 3})^{2} \geq 0

Answer: (A)

The feasible region requires

x > - 7/2

x \neq = - 5/2

x \neq = 3/2

x \neq = 7

. For

lo g_{a} b \geq 0

, either (i)

a > 1

and

b \geq 1

, or (ii)

0 < a < 1

and

0 < b \leq 1

. Case (i):

x + 7/2 > 1

and

(\frac{x - 7}{2 x - 3})^{2} \geq 1

gives

x > - 5/2

and

(3 x - 10) (x + 4) \leq 0

, so

x \in (- 5/2, 10/3]

. Case (ii) yields no common values. Excluding

x = 3/2

, integral solutions are

{- 2, - 1, 0, 1, 2, 3}

, giving 6 solutions.

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option