$\left\{x \in R: \frac{14 x}{x+1}-\frac{9 x-30}{x-4} < 0\right\}$ is equal to

• A

  $(-1,4)$
• B

  $(1,4) \cup(5,7)$
• C

  $(1,7)$
• Correct answer
  D

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

Solution

$\left\{x \in R: \frac{14 x}{x+1}-\frac{9 x-30}{x-4} < 0\right\}$ $\frac{14 x(x-4)-(9 x-30)(x+1)}{(x+1)(x-4)} < 0$ $\frac{14 x^2-56 x-\left(9 x^2-30 x+9 x-30\right)}{(x+1)(x-4)} < 0$ $\frac{\left(14 x^2-56 x-9 x^2+30 x-9 x+30\right)}{(x+1)(x-4)} < 0$ $\frac{\left(5 x^2-35 x+30\right)}{(x+1)(x-4)} < 0$ $\frac{\left(x^2-7 x+6\right)}{(x+1)(x-4)} < 0, \frac{(x-1)(x-6)}{(x+1)(x-4)} < 0$ Drawn number line,
Hence, $x \in(-1,1) \cup(4,6)$

Hence, the correct answer is option: 'D'.