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