The number of integral solutions x of log(x+72) (x−72x−3)2≥0 is

For log(x+72) (x−72x−3)2to be defined x+72>0 and x+72≠1⇒ x>−72>0 and x≠−52Case - I : −72 −52log(x+72) (x−72x−3)2≥0⇒ (x−72x−3)2≥1 ⇒ 3x2+2x−40(2x−3)2≤0 ⇒ (3x−10) (x+4)(2x−3)2≤0∴ x∈[−4, 103] and x≥−52Integral values of x∈{−2, −1, 0, 1, 2, 3}∴No. of integral values of x=6

The number of integral solutions x of log(x+72) (x−72x−3)2≥0 is

For log(x+72) (x−72x−3)2to be defined x+72>0 and x+72≠1⇒ x>−72>0 and x≠−52Case - I : −72 −52log(x+72) (x−72x−3)2≥0⇒ (x−72x−3)2≥1 ⇒ 3x2+2x−40(2x−3)2≤0 ⇒ (3x−10) (x+4)(2x−3)2≤0∴ x∈[−4, 103] and x≥−52Integral values of x∈{−2, −1, 0, 1, 2, 3}∴No. of integral values of x=6

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

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Detailed Solution

For $\log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2}$ to be defined $x + \frac{7}{2} > 0 and x + \frac{7}{2} \neq 1$
$\Rightarrow x > - \frac{7}{2} > 0 and x \neq - \frac{5}{2}$
Case - I : $- \frac{7}{2} < x < \frac{- 5}{2} \log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2} \geq 0 \Rightarrow {(\frac{x - 7}{2 x - 3})}^{2} \leq 1$
$\Rightarrow \frac{{(x - 7)}^{2}}{{(2 x - 3)}^{2}} - 1 \leq 0 \Rightarrow \frac{- 3 x^{2} - 2 x + 40}{{(2 x - 3)}^{2}} \leq 0 \Rightarrow \frac{3 x^{2} + 2 x - 40}{{(2 x - 3)}^{2}} \leq 0$
$\Rightarrow \frac{(3 x - 10) (x + 4)}{{(2 x - 3)}^{2}} \geq 0 \Rightarrow x \leq - 4 or x \geq \frac{10}{3} ∴ x \in ϕ$
Case - II : $x > - \frac{5}{2} \log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2} \geq 0 \Rightarrow {(\frac{x - 7}{2 x - 3})}^{2} \geq 1 \Rightarrow \frac{3 x^{2} + 2 x - 40}{{(2 x - 3)}^{2}} \leq 0 \Rightarrow \frac{(3 x - 10) (x + 4)}{{(2 x - 3)}^{2}} \leq 0 ∴ x \in [- 4, \frac{10}{3}] and x \geq - \frac{5}{2}$
Integral values of $x \in {- 2, - 1, 0, 1, 2, 3}$
$∴$ No. of integral values of $x = 6$