# What are the boundaries of x if (2x-1)/(x+5) >= (x+2)/(x+3)?

Mar 2, 2018

$x = - 5 , x = - 3 , x = 1 - \sqrt{14} , x = 1 + \sqrt{14}$
$\ge \text{ occurs for "x <-5" and "x>=1+sqrt(14)" and}$
$- 3 < x \le 1 - \sqrt{14} \text{.}$

#### Explanation:

$\implies \frac{2 x - 1}{x + 5} - \frac{x + 2}{x + 3} \ge 0$

$\implies \frac{\left(2 x - 1\right) \left(x + 3\right) - \left(x + 2\right) \left(x + 5\right)}{\left(x + 5\right) \left(x + 3\right)} \ge 0$

$\implies \frac{2 {x}^{2} + 5 x - 3 - {x}^{2} - 7 x - 10}{\left(x + 5\right) \left(x + 3\right)} \ge 0$

$\implies \frac{{x}^{2} - 2 x - 13}{\left(x + 5\right) \left(x + 3\right)} \ge 0$

$\implies \frac{\left(x - 1 - \sqrt{14}\right) \left(x - 1 + \sqrt{14}\right)}{\left(x + 5\right) \left(x + 3\right)} \ge 0$

$\text{We have following zeros in order of magnitude : }$

$\ldots . - 5 \ldots . - 3 \ldots . 1 - \sqrt{14} \ldots .1 + \sqrt{14} \ldots . .$
$- - - - - - - - - - - 0 + + +$
$- - - - - - - 0 + + + + + + +$
$- - - - - 0 + + + + + + + + +$
$- - 0 + + + + + + + + + + + +$
$\text{=========================}$
$+ + 0 - - - 0 + + 0 - - - 0 + + +$

$\text{We see ">=0" occurs for "x <-5" and "x>=1+sqrt(14)" and}$
$- 3 < x \le 1 - \sqrt{14} \text{.}$