Question

Solution of the inequation

$\log_{0.1}\left( \log_{2}\frac{x^{2} + 1}{|x - 1|} \right)$< 0 contains the interval

• A. (1, ¥)
• B. (–¥, 1)
• C. [1, ¥)
• D. None of these

Answer 1

Answer: (A)

(1, ¥)

Answer 2

log₂ $\left( \frac{x^{2} + 1}{|x - 1|} \right)$ > 1̃$\frac{x^{2} + 1}{|x - 1|}$ > 2

If x > 1 x² + 1 > 2 |x – 1|

x² + 1 – 2x + 2 > 0 " xÎR

If x < 1 x² + 1 > 2 – 2x

x² + 2x + 1 > 2

̃x + 1 $> \sqrt{2}$ ̃ x >$\sqrt{2}$– 1 and x +1<–$\sqrt{2}$

̃ x < – 1 – $\sqrt{2}$

thus**,** xÎ (– ¥, –1 –$\sqrt{2}$) È $(\sqrt{2} - 1,1)$ È (1, ¥)