The sets of real values of ' $x$ ' for which log ${log}_{2 x + 3} x^{2} < {log}_{2 x + 3} (2 x + 3) (1)$ would include which of the given points?

(- \frac{3}{2}, 1)

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(- 1, 0)

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(0, 3)

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(- \infty, 0)

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Solution

The correct options are
A $(- \frac{3}{2}, 1)$
B $(- 1, 0)$
C $(0, 3)$
The inequality (1) is equivalent to the following system of inequalities
$0 < 2 x + 3 < 1$
$x^{2} > 2 x + 3$ ..............(1)
or $2 x + 3 > 1$
$0 < x^{2} < 2 x + 3$ ............(2)
Solving (1) first, we see that it is equivalent to
$- 3 / 2 < x < - 1$ and $(x - 3) (x + 1) > 0$
$\Rightarrow - 3 / 2 < x < - 1$ and $(x < - 1$ or $x > 3)$
$\Rightarrow - 3 / 2 < x < - 1$
Next, solving (2), we see that it is equivalent to
$x > - 1$ and $(x + 1) (x - 3) < 0 \Rightarrow - 1 < x < 3$
Also, $x \neq 0$
Hence, $x \in (- 3 / 2, - 1) ⋃ (- 1, 0) ⋃ (0, 3)$

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