To solve the rational inequality:

$\frac{2x - 8}{x - 2} \geq 0$

we will follow these steps:

Step 1: Simplify the expression

The numerator of the rational expression is $2x - 8$, which can be factored as:

$2x - 8 = 2(x - 4)$

So, the inequality becomes:

$\frac{2(x - 4)}{x - 2} \geq 0$

Step 2: Find critical points

Critical points occur where the numerator or denominator is zero, because these points divide the real number line into intervals where the sign of the expression may change.

• Numerator: $2(x - 4) = 0$ when $x = 4$
• Denominator: $x - 2 = 0$ when $x = 2$

Thus, the critical points are $x = 4$ and $x = 2$.

Step 3: Analyze intervals

The critical points $x = 2$ and $x = 4$ divide the real number line into three intervals:

1. $(-\infty, 2)$
2. $(2, 4)$
3. $(4, \infty)$

We need to determine the sign of the expression $\frac{2(x - 4)}{x - 2}$ in each interval.

• For $x \in (-\infty, 2)$: Choose $x = 0$. $\frac{2(0 - 4)}{0 - 2} = \frac{2(-4)}{-2} = 4 > 0$ The expression is positive in this interval.

• For $x \in (2, 4)$: Choose $x = 3$. $\frac{2(3 - 4)}{3 - 2} = \frac{2(-1)}{1} = -2 < 0$ The expression is negative in this interval.

• For $x \in (4, \infty)$: Choose $x = 5$. $\frac{2(5 - 4)}{5 - 2} = \frac{2(1)}{3} = \frac{2}{3} > 0$ The expression is positive in this interval.

Step 4: Include boundary points

Now we need to consider whether the critical points $x = 2$ and $x = 4$ satisfy the inequality.

• At $x = 4$, the expression equals zero: $\frac{2(4 - 4)}{4 - 2} = 0$ Since the inequality is $\geq 0$, $x = 4$ is included in the solution.

• At $x = 2$, the expression is undefined because the denominator is zero. Therefore, $x = 2$ is not included in the solution.

Step 5: Write the solution

The solution to the inequality is the set of values where the expression is non-negative. From our interval analysis:

$x \in (-\infty, 2) \cup [4, \infty)$

Thus, the solution is:

$x \in (-\infty, 2) \cup [4, \infty)$

Would you like more details on any part of the solution?

Here are 5 related questions to expand on this topic:

1. How do you solve rational inequalities with more complicated expressions in the numerator and denominator?
2. Why is $x = 2$ excluded from the solution set in this problem?
3. How would the solution change if the inequality was strict (i.e., $>$ instead of $\geq$)?
4. Can you explain how the sign of a rational expression changes at critical points?
5. What are some real-world applications of solving rational inequalities?

Tip: Always check the sign of the rational expression in different intervals formed by the critical points to determine the solution.