# How do you solve (2t+7)/(t-4)>=3 using a sign chart?

##### 1 Answer
Feb 9, 2017

The solution is t in ]4, 19]

#### Explanation:

Rewrite the equation

$\frac{2 t + 7}{t - 4} \ge 3$

$\frac{2 t + 7}{t - 4} - 3 \ge 0$

$\frac{\left(2 t + 7\right) - 3 \left(t - 4\right)}{t - 4} \ge 0$

$\frac{2 t + 7 - 3 t + 12}{t - 4} \ge 0$

$\frac{- t + 19}{t - 4} \ge 0$

Let $f \left(t\right) = \frac{- t + 19}{t - 4}$

We can construct the sign chart

$\textcolor{w h i t e}{a a a a}$$t$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$4$$\textcolor{w h i t e}{a a a a a a a a}$$19$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$19 - t$$\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}$$+$$\textcolor{w h i t e}{a a a a a}$$-$

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

Therefore,

$f \left(t\right) \ge 0$ when t in ]4, 19]