x∈ℝ:14xx+1−9x−30x−4<0=

14xx+1−9x−30x−4<0⇒14x(x−4)−(9x−30)(x+1)(x+1)(x−4)<0⇒14x2−56x−9x2+21x+30(x+1)(x−4)<0⇒5x2−35x+30(x+1)(x−4)<0⇒5(x−1)(x−6)(x+1)(x−4)<0⇒5(x+1)(x−1)(x−4)(x−6)<0⇒−1<x<1,4<x<6Given set=(−1,1)∪(4,6)

x∈ℝ:14xx+1−9x−30x−4<0=

14xx+1−9x−30x−4<0⇒14x(x−4)−(9x−30)(x+1)(x+1)(x−4)<0⇒14x2−56x−9x2+21x+30(x+1)(x−4)<0⇒5x2−35x+30(x+1)(x−4)<0⇒5(x−1)(x−6)(x+1)(x−4)<0⇒5(x+1)(x−1)(x−4)(x−6)<0⇒−1<x<1,4<x<6Given set=(−1,1)∪(4,6)

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$\{x \in ℝ : \frac{14 x}{x + 1} - \frac{9 x - 30}{x - 4} < 0\} =$

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$(- 1, 4)$

$(1, 4) \cup (5, 7)$

$(1, 7)$

$(- 1, 1) \cup (4, 6)$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{14 x}{x + 1} - \frac{9 x - 30}{x - 4} < 0 \Rightarrow \frac{14 x (x - 4) - (9 x - 30) (x + 1)}{(x + 1) (x - 4)} < 0 \Rightarrow \frac{14 x^{2} - 56 x - 9 x^{2} + 21 x + 30}{(x + 1) (x - 4)} < 0 \\ \Rightarrow \frac{5 x^{2} - 35 x + 30}{(x + 1) (x - 4)} < 0 \Rightarrow \frac{5 (x - 1) (x - 6)}{(x + 1) (x - 4)} < 0 \\ \Rightarrow 5 (x + 1) (x - 1) (x - 4) (x - 6) < 0 \Rightarrow - 1 < x < 1, 4 < x < 6 \end{array}$