How do you solve  5/(x+1) - 1/2 = 2/3x + 3?

Nov 10, 2017

$x = \frac{1}{8} \left(- 25 + \sqrt{385}\right)$ or $x = \frac{1}{8} \left(- 25 - \sqrt{385}\right)$

Explanation:

We will first write each side as a single fraction; then multiply this out and solve the quadratic equation that follows.

$\frac{5}{x + 1} - \frac{1}{2} = \frac{2 x}{3} + 3$
$\frac{10}{2 \left(x + 1\right)} - \frac{x + 1}{2 \left(x + 1\right)} = \frac{2 x}{3} + \frac{9}{3}$
$\frac{11 - x}{2 x + 2} = \frac{2 x + 9}{3}$

Here we cross-multiply, to get:

$3 \left(11 - x\right) = \left(2 x + 2\right) \left(2 x + 9\right)$
$33 - 3 x = 4 {x}^{2} + 22 x + 18$
$4 {x}^{2} + 25 x - 15 = 0$

You can now use The Formula or complete the square; I'd recommend The Formula.

$x = \frac{- 25 \pm \sqrt{{25}^{2} - 4 \times 4 \times \left(- 15\right)}}{2 \times 4}$

$x = \frac{- 25 \pm \sqrt{625 - 240}}{8}$
$x = \frac{- 25 \pm \sqrt{385}}{8}$

$\therefore x = \frac{1}{8} \left(- 25 + \sqrt{385}\right)$ or $x = \frac{1}{8} \left(- 25 - \sqrt{385}\right)$
(x~~-0.672, x~~-5.58)