Question

Let T = `{x | (x + 5)/(x - 7) - 5 = (4x - 40)/(13 - x)}`. Is T an empty set? Justify your answer.

Answer 1

Given: T = `{x | (x + 5)/(x - 7) - 5 = (4x - 40)/(13 - x)}`

To check: T is an empty set or not

`(x + 5)/(x - 7) - 5 = (4x - 40)/(13 - x)`

⇒ `(x + 5 - 5(x - 7))/(x - 7) = (4x - 40)/(13 - x)`

⇒ `(x + 5 - 5x + 35)/(x - 7) = (4x - 40)/(13 - x)`

⇒ `(-4x + 40)/(x - 7) = (4x - 40)/(13 - x)`

⇒ `(-(4x - 40))/(x - 7) = (4x - 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