# How do solve 2/(x+1)>=1/(x-2) and write the answer as a inequality and interval notation?

Jan 7, 2017

$- 1 < x < 2 \mathmr{and} x \ge 5$

or, in interval notation:

]-1;2[uu[5;oo[

#### Explanation:

The inequality can be rewritten as:

$\frac{2}{x + 1} - \frac{1}{x - 2} \ge 0$

and then

$\frac{2 \left(x - 2\right) - \left(x + 1\right)}{\left(x + 1\right) \left(x - 2\right)} \ge 0$

$\frac{2 x - 4 - x - 1}{\left(x + 1\right) \left(x - 2\right)} \ge 0$

$\frac{x - 5}{\left(x + 1\right) \left(x - 2\right)} \ge 0$

Let's study the sign of each binomial:

1) $x - 5 \ge 0 \to x \ge 5$
2) $x + 1 > 0 \to x > - 1$
3) $x - 2 > 0 \to x > 2$

$$  __-1__2__5__


1) $-$ $-$$-$o$+$
2)$-$ $+$ $+$ $+$
3)$-$ $-$ $+$ $+$

Then the inequality is verified when the fraction is greater than 0, that's:

$- 1 < x < 2 \mathmr{and} x \ge 5$

or, in interval notation:

]-1;2[uu[5;oo[