What is x/((x+2))<= 1/x ?

The solution to x/((x+2))<= 1/x is -2<x<=-1\lor 0<x<= 2

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x/((x+2))<= 1/x

\frac{x}{( x + 2 )} \leq \frac{1}{x}

- 2 < x \leq - 1 or 0 < x \leq 2

Interval Notation

(- 2, - 1] \cup (0, 2]

Solution steps

\frac{x}{x + 2} \leq \frac{1}{x}

Rewrite in standard form

\frac{x ^{2} - x - 2}{x ( x + 2 )} \leq 0

Factor

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

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

Identify the intervals

- 2 < x \leq - 1 or 0 < x \leq 2

\frac{( x ^{2} - 9 )}{( 3 x - x ^{2} - 24 )} < 0

2 x^{2} - 2 x - 1 < 0

\frac{( 2 x )}{( 2 x ^{2} + 5 x + 2 )} > \frac{1}{( x + 1 )}

\frac{1}{2 ( 3 - 8 x )} > 20 (\frac{1 - 1}{5 x})

\frac{1}{( x - 2 )} \leq 0

What is x/((x+2))<= 1/x ?
The solution to x/((x+2))<= 1/x is -2<x<=-1\lor 0<x<= 2