https://file.aiquickdraw.com/mathbot/file/1726588848519asg81pns.jpeg

Let's analyze the two problems in the image:

### Problem 1:

\[
\frac{2x - 8}{x - 2} \geq 0
\]

This is an inequality involving a rational expression. To solve it, we will:
1. Find the values of \( x \) that make the numerator and denominator zero.
2. Analyze the sign of the expression in different intervals of \( x \) using these critical points.

Steps:
- The numerator \( 2x - 8 = 0 \) gives \( x = 4 \).
- The denominator \( x - 2 = 0 \) gives \( x = 2 \).

These points divide the real number line into intervals. We will test each interval to determine where the inequality holds.

### Problem 2:

\[
\frac{x}{x-2} - 7 = \frac{2}{x-2}
\]

This is an equation involving fractions with a common denominator. To solve:
1. Combine the terms on the left-hand side into a single fraction.
2. Simplify and solve for \( x \).

Would you like to go through the detailed solution for each problem? Let me know!

Here are 5 relative questions for further exploration:
1. How do you handle inequalities with rational expressions in general?
2. What happens if the denominator of a rational inequality is zero?
3. How do you determine if a fraction is positive or negative?
4. How do you combine and simplify algebraic fractions?
5. Can a rational equation have no solution? Why?

**Tip:** When solving rational inequalities, always check for undefined points (when the denominator is zero) and exclude them from your solution set.

Solve the following: 1) (2x - 8)/(x - 2) ≥ 0 and 2) x/(x - 2) - 7 = 2/(x - 2)

Learn how to solve a rational inequality (2x - 8)/(x - 2) ≥ 0 and an equation involving fractions. This problem covers key algebraic concepts such as inequality analysis and rational equation solving. Suitable for high school students in Grades 10-12.

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