Question:
Let,
$T=\left\{x \mid \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right\}$
Is T an empty set? Justify your answer.
Solution:
According to the question,
$T=\left\{x \mid \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right\}$
To check whether T is an empty set or not,
We solve,
$\frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}$
$\Rightarrow \frac{x+5-5(x-7)}{x-7}=\frac{4 x-40}{13-x}$
$\Rightarrow \frac{x+5-5 x+35}{x-7}=\frac{4 x-40}{13-x}$
$\Rightarrow \frac{-4 x+40}{x-7}=\frac{4 x-40}{13-x}$
$\Rightarrow \frac{-(4 x-40)}{x-7}=\frac{4 x-40}{13-x}$
⇒ -(4x – 40)(13 – x) = (4x – 40)(x – 7)
⇒ (4x – 40)(x – 7) + (4x – 40)(13 – x) = 0
⇒ (4x – 40)(x – 7 + 13 – x) = 0
⇒ 6(4x – 40) = 0
⇒ 24(x – 10) = 0
⇒ x – 10 = 0
⇒ x = 10
So, T = {10}
⇒ T is not an empty set