What is x/(x+2)> 1/(x-3) ?

The solution to x/(x+2)> 1/(x-3) is x 2+sqrt(6)

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x/(x+2)> 1/(x-3)

\frac{x}{x + 2} > \frac{1}{x - 3}

x < - 2 or 2 - 6 < x < 3 or x > 2 + 6

Interval Notation

(- \infty, - 2) \cup (2 - 6, 3) \cup (2 + 6, \infty)

Decimal Notation

x < - 2 or - 0.44948 \dots < x < 3 or x > 4.44948 \dots

Solution steps

\frac{x}{x + 2} > \frac{1}{x - 3}

Rewrite in standard form

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

Factor

x^{2} - 4 x - 2 : (x - 2 - 6) (x - 2 + 6)

\frac{( x - 2 - 6 ) ( x - 2 + 6 )}{( x + 2 ) ( x - 3 )} > 0

Identify the intervals

x < - 2 or 2 - 6 < x < 3 or x > 2 + 6

\frac{- 1}{( x + 2 )} > 0

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

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

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

(x^{2} - 2 x) (3 x - 6) \leq 0

What is x/(x+2)> 1/(x-3) ?
The solution to x/(x+2)> 1/(x-3) is x<-2\lor 2-sqrt(6)<x<3\lor x>2+sqrt(6)