# How do solve 3/(x-3)<=2/(x+2) and write the answer as a inequality and interval notation?

Jun 13, 2017

Inequality: $x \le - 12$ and $- 2 < x < 3$
Interval: $\left(- \infty , - 12\right]$ and $\left(- 2 , 3\right)$

#### Explanation:

Step 1. Find the critical values by assuming equality.

Assume $\frac{3}{x - 3} = \frac{2}{x + 2}$

You have critical values at $x = 3$ and $x = - 2$ because these would cause the equation to divide by zero.

Also, solving for $x$ gives the last critical value

$3 \left(x + 2\right) = 2 \left(x - 3\right)$

$3 x + 6 = 2 x - 6$

$x = - 12$

Step 2. Evaluate the inequality around these critical values.

{:("Crit. Value ","Test value ", 3/(x-3) <= 2/(x+2)),(x <= -12 ," "-20," True"),(-12 < x < -2, " "-10, " False"),(-2 < x < 3," "0," True"),(x > 3, " "5," False"):}

Step 3. Complete by writing the inequality and interval notation.

Inequality: $\text{ } x \le - 12$ and $- 2 < x < 3$
Interval: $\text{ } \left(- \infty , - 12\right]$ and $\left(- 2 , 3\right)$