# How do solve (3x+1)/(x+4)<=1 and write the answer as a inequality and interval notation?

Mar 10, 2018

The solution is $x \in \left(- 4 , \frac{3}{2}\right]$ or $- 4 < x \le \frac{3}{2}$

#### Explanation:

Let's rewrite and simplify the inequality

$\frac{3 x + 1}{x + 4} \le 1$

$\frac{3 x + 1}{x + 4} - 1 \le 0$

$\frac{\left(3 x + 1\right) - \left(x + 4\right)}{x + 4} \le 0$

$\frac{\left(2 x - 3\right)}{x + 4} \le 0$

Let $f \left(x\right) = \frac{\left(2 x - 3\right)}{x + 4}$

Now, we can build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a a a}$$\frac{3}{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 4$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$2 x - 3$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a}$color(white)(aaaaa)-$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- 4 , \frac{3}{2}\right]$ or $- 4 < x \le \frac{3}{2}$

graph{(3x+1)/(x+4)-1 [-27.09, 18.51, -12.22, 10.59]}