Question

Solve each inequality. Graph the solution set.

$\frac{6}{x+3}\ge 1$

Short answer

The real zeroes are 3 and $-3$and the solution set can be graphed as

Step-by-step solution

Step 1. Given Information

The given inequality is$\frac{6}{x+3}\ge 1$

Step 2. Calculation

Solve the given inequality as follows,

$$\begin{gathered} \frac{6}{x+3}-1\ge 1-1 \\ \frac{6}{x+3}-1\ge 0 \\ \frac{6}{x+3}-1\left(\frac{x+3}{x+3}\right)\ge 0 \\ \frac{-x+3}{x+3}\ge 0 \end{gathered}$$

Now, let $f\left(x\right)=\frac{-x+3}{x+3}$. We will find the real zeroes of the numerator and denominator.

$$\begin{gathered} -x+3=0\Rightarrow x=3 \\ x+3=0\Rightarrow x=-3 \end{gathered}$$

Thus, real zeroes of the function are$-3and3$

Step 3. Graph 

Construct the table by using a number from each interval to check whether the function is positive or negative.

Thus, the solution set is $\left\{x|-3<x\le 3\right\}$

And the solution set can be graphed as