The solution set of the equation, (x+3)log10⁡x(x−4)x2+9>0

Domain; x>0 and x≠4; x2+9>0 ∴(x+3)log10xx−4>0⇒log10 xx−4>0x∈(0,1)⇒log10x 0

The solution set of the equation, (x+3)log10⁡x(x−4)x2+9>0

Domain; x>0 and x≠4; x2+9>0 ∴(x+3)log10xx−4>0⇒log10 xx−4>0x∈(0,1)⇒log10x 0

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The solution set of the equation, $\frac{(x + 3) \log_{10} x}{(x - 4) (x^{2} + 9)} > 0$

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$(0, 1) \cup (4, \infty)$

$(0, 1) \cup [4, \infty)$

$(0, 2) \cup [4, \infty)$

$(0, 1] \cup [4, \infty)$

answer is D.

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Detailed Solution

$Domain; x > 0 and x \neq 4;$
$x^{2} + 9 > 0 ∴ \frac{(x + 3) \log_{10}^{x}}{x - 4} > 0$
$\Rightarrow \frac{\log_{10}^{x}}{x - 4} > 0$
$x \in (0, 1) \Rightarrow \log_{10}^{x} < 0; x - 4 < 0 \Rightarrow \frac{\log_{10}^{x}}{x - 4} > 0$

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The solution set of the equation, (x+3)log10⁡x(x−4)x2+9>0

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The solution set of the equation, $\frac{(x + 3) \log_{10} x}{(x - 4) (x^{2} + 9)} > 0$