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  • Let Is T an empty set? Justify your answer.

Let

T = \left\{x \ | \ \frac{x+5}{x-7} - 5 = \frac{4x-40}{13-x} \right \}

Is T an empty set? Justify your answer.
 

Answers (1)

Given: T = \left\{x \ | \ \frac{x+5}{x-7} - 5 = \frac{4x-40}{13-x} \right \}

                \begin{aligned}\Rightarrow& & \frac{x+5}{x-7}- 5 &= \frac{4x-40}{13-x} \\ && \frac{(x+5)-5(x-7)}{x-7} & = \frac{4x-40}{13-x} \\ \Rightarrow && \frac{x + 5 - 5x + 35}{x-7} & = \frac{4x-40}{13-x} \\ \Rightarrow & & -4(x-10)(13-x) &= 4( x -10)(x-7) \\ \Rightarrow & & -4(x-10) (13-x + x -7) & = 0 \\ \Rightarrow & & -4(x - 10)6 &= 0 \\ \Rightarrow & & -24(x-10) &=0 \\ \Rightarrow & & x - 10 &= 0 \\ \Rightarrow & & x = 10& \text{ or }T = \{10\}\end{aligned}

Hence T is not an empty set.

Posted by

infoexpert22

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