What is ((x+4))/(x-2)-((x+3))/(x-5)>0 ?

The solution to ((x+4))/(x-2)-((x+3))/(x-5)>0 is x<-7\lor 2<x<5

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((x+4))/(x-2)-((x+3))/(x-5)>0

\frac{( x + 4 )}{x - 2} - \frac{( x + 3 )}{x - 5} > 0

x < - 7 or 2 < x < 5

Interval Notation

(- \infty, - 7) \cup (2, 5)

Solution steps

\frac{x + 4}{x - 2} - \frac{x + 3}{x - 5} > 0

Rewrite in standard form

\frac{- 2 x - 14}{( x - 2 ) ( x - 5 )} > 0

Factor

\frac{- 2 x - 14}{( x - 2 ) ( x - 5 )} : \frac{- 2 ( x + 7 )}{( x - 2 ) ( x - 5 )}

\frac{- 2 ( x + 7 )}{( x - 2 ) ( x - 5 )} > 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 2 ( x + 7 ) ) ( - 1 )}{( x - 2 ) ( x - 5 )} < 0 \cdot (- 1)

Simplify

\frac{2 ( x + 7 )}{( x - 2 ) ( x - 5 )} < 0

Divide both sides by

2

\frac{\frac{2 ( x + 7 )}{( x - 2 ) ( x - 5 )}}{2} < \frac{0}{2}

Simplify

\frac{x + 7}{( x - 2 ) ( x - 5 )} < 0

Identify the intervals

x < - 7 or 2 < x < 5

4^{2} + 7^{2}

- (x + 2) < - 2 (- 2 x + 5)

1 6^{2} - 25

x^{2} - 4 x + 3 \geq 0

(- 3 + 2 i) (1 - i^{3})

What is ((x+4))/(x-2)-((x+3))/(x-5)>0 ?
The solution to ((x+4))/(x-2)-((x+3))/(x-5)>0 is x<-7\lor 2<x<5