Question

Solution of inequality

$\log_{\log_{2}\left( \frac{x}{2} \right)}{}$ (x² – 10x + 22) > 0 is –

• A. (– ¥, 3)
• B. (5 –$\sqrt{3}$, 3) È (5 +$\sqrt{3}$, 7)
• C. (0, 5 –$\sqrt{3}$) È (3, 4)
• D. (7, ¥)

Answer 1

Answer: (D)

(7, ¥)

Answer 2

Inequality $\log_{\log_{2}\left( \frac{x}{2} \right)}{}$ (x² – 10x + 22) > 0 …(1)

L.H.S. is valid if :

x² – 10x + 22 > 0 $\frac{x}{2} > 0$

x < 5 – $\sqrt{3}$ or x > 5 + $\sqrt{3}$ x > 0

eqⁿ (1) will be solved for two cases

(1) 0 < log₂$\left( \frac{x}{2} \right)$ < 1

̃ 1 < $\frac{x}{2}$ < 2 = ̃ 2 < x < 4

$\log_{\log_{2}\left( \frac{x}{2} \right)}{}$ (x² – 10x + 22) > 0

x² – 10x + 22 < 1

x² – 10x + 21 < 0 ̃ 3 < x < 7

The common solution 3 < x < 4

(2) log₂$\left( \frac{x}{2} \right)$ > 1 ̃ $\frac{x}{2}$ > 2

x > 4

$\log_{\log_{2}\left( \frac{x}{2} \right)}{}$ (x² – 10x + 22) > 0

x² – 10x + 22 > 1 ̃ x² – 10x + 21 > 0

x < 3 or x > 7 common solⁿ x > 7

two cases x Î (3, 4) È (7, ¥)

Now common solution with initial values

x Î (7, ¥)