Question

Solve each inequality algebraically.

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

Short answer

Solution to the inequality $\frac{x+2}{x-4}\ge 1$isrole="math" localid="1646128818261" $(4,\infty )$.

Step-by-step solution

Step 1. Given information 

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

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 

$$\begin{gathered} \frac{x+2}{x-4}\ge 1 \\ \frac{x+2}{x-4}-1\ge 1-1(\mathrm{Subtract}1\mathrm{from}\mathrm{both}\mathrm{sides}) \\ \frac{\mathrm{x}+2-1(\mathrm{x}-4)}{\mathrm{x}-4}\ge 0(\mathrm{Take}\mathrm{LCD}) \\ \frac{\mathrm{x}+2-\mathrm{x}+4}{\mathrm{x}-4}\ge 0 \\ \frac{6}{\mathrm{x}-4}\ge 0 \end{gathered}$$

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

Assume $f\left(x\right)=\frac{6}{x-4}$.

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

Step 4. Form the intervals 

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

$(-\infty ,4)\cup (4,\infty )$

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

Create the following table:

Interval	$(-\infty ,4)$	$(4,\infty )$
Number chosen	0	5
Value of f	$f\left(0\right)=-\frac{3}{2}$	$f\left(5\right)=6$
Conclusion	negative	positive

Step 6. Identify the interval

Since we want to know where f is positive or zero, we conclude that $f\left(x\right)\ge 0$in the intervalrole="math" localid="1646129255698" $(4,\infty )$.