Question

Solve each inequality algebraically.

$\frac{x-4}{2x+4}\ge 1$

Short answer

Solution to the inequality$\frac{x-4}{2x+4}\ge 1$ is $[-8,-2)$.

Step-by-step solution

Step 1. Given information  

We have been given an inequality $\frac{x-4}{2x+4}\ge 1$.

We have to solve this inequality algebraically.

Step 2. Write the inequality so that a polynomial or rational expression f is on the left side and zero is on the right side  

$$\begin{gathered} \frac{x-4}{2x+4}\ge 1 \\ \frac{x-4}{2x+4}-1\ge 1-1 \\ \frac{x-4-1(2x+4)}{2x+4}\ge 0 \\ \frac{x-4-2x-4}{2x+4}\ge 0 \\ \frac{-x-8}{2x+4}\ge 0 \\ \frac{x+8}{2x+4}\le 0 \end{gathered}$$

Step 3. Determine the real numbers at which the expression f equals zero and at which the expression f is undefined. 

Assume $f\left(x\right)=\frac{x+8}{2x+4}$.

$$\begin{gathered} x+8=0 \\ x=-8 \\ 2x+4=0 \\ 2x=-4 \\ x=-2 \end{gathered}$$

Step 4. Form the intervals  

Using the values of x found in previous step, we can divide the real numbers in the intervals:

$(-\infty ,-8)\cup (-8,-2)\cup (-2,\infty )$

Step 5. Select a number in each interval and evaluate f at the number   

Create the following table:

Interval	$(-\infty ,-8)$	$(-8,-2)$	$(-2,\infty )$
Number chosen	$-9$	$-3$	0
Value of f	$f(-9)=\frac{1}{14}$	$f(-3)=-\frac{5}{2}$	$f\left(0\right)=2$
Conclusion	positive	negative	positive

Step 6. Identify the interval 

Since we want to know where f is negative or zero, we conclude that $f\left(x\right)\le 0$ in the interval $[-8,-2)$.