What is ((14x))/((x+1))<((9x-30))/((x-4)) ?

The solution to ((14x))/((x+1))<((9x-30))/((x-4)) is -1<x<1\lor 4<x<6

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((14x))/((x+1))<((9x-30))/((x-4))

\frac{( 14 x )}{( x + 1 )} < \frac{( 9 x - 30 )}{( x - 4 )}

- 1 < x < 1 or 4 < x < 6

Interval Notation

(- 1, 1) \cup (4, 6)

Solution steps

\frac{14 x}{x + 1} < \frac{9 x - 30}{x - 4}

Rewrite in standard form

\frac{5 x ^{2} - 35 x + 30}{( x + 1 ) ( x - 4 )} < 0

Factor

\frac{5 x ^{2} - 35 x + 30}{( x + 1 ) ( x - 4 )} : \frac{5 ( x - 1 ) ( x - 6 )}{( x + 1 ) ( x - 4 )}

\frac{5 ( x - 1 ) ( x - 6 )}{( x + 1 ) ( x - 4 )} < 0

Divide both sides by

5

\frac{\frac{5 ( x - 1 ) ( x - 6 )}{( x + 1 ) ( x - 4 )}}{5} < \frac{0}{5}

Simplify

\frac{( x - 1 ) ( x - 6 )}{( x + 1 ) ( x - 4 )} < 0

Identify the intervals

- 1 < x < 1 or 4 < x < 6

x^{5} - 2 x^{4} - x^{3} - 2 x^{2} - 20 x + 24 \leq 0

a (a - 2) \geq 2 (a + 6)

\frac{x}{( x - 1 )} > \frac{( x - 1 )}{x}

\frac{20 ( x - 2 )}{( x ^{2} + 1 )} < \frac{- 1}{2}

x^{2} (x - 2) (x - 3) (1 - x) (1 + x) \geq 0

What is ((14x))/((x+1))<((9x-30))/((x-4)) ?
The solution to ((14x))/((x+1))<((9x-30))/((x-4)) is -1<x<1\lor 4<x<6