What is ((x-2)}{(x+3)}<\frac{(x+1))/x ?

The solution to ((x-2)}{(x+3)} 0

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

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

- 3 < x < - \frac{1}{2} or x > 0

Interval Notation

(- 3, - \frac{1}{2}) \cup (0, \infty)

Decimal Notation

- 3 < x < - 0.5 or x > 0

Solution steps

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

Rewrite in standard form

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

Factor

\frac{- 6 x - 3}{x ( x + 3 )} : \frac{- 3 ( 2 x + 1 )}{x ( x + 3 )}

\frac{- 3 ( 2 x + 1 )}{x ( x + 3 )} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 3 ( 2 x + 1 ) ) ( - 1 )}{x ( x + 3 )} > 0 \cdot (- 1)

Simplify

\frac{3 ( 2 x + 1 )}{x ( x + 3 )} > 0

Divide both sides by

3

\frac{\frac{3 ( 2 x + 1 )}{x ( x + 3 )}}{3} > \frac{0}{3}

Simplify

\frac{2 x + 1}{x ( x + 3 )} > 0

Identify the intervals

- 3 < x < - \frac{1}{2} or x > 0

(x - 1) (x - 2) (x - 3) > 0

(x - 1)^{2} (x + 1)^{3} (x - 4) < 0

- x^{2} + x + 1 \leq 0

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

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

What is ((x-2)}{(x+3)}<\frac{(x+1))/x ?
The solution to ((x-2)}{(x+3)}<\frac{(x+1))/x is -3<x<-1/2 \lor x>0