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The image you uploaded contains two rational inequalities. Let's solve them one by one.

### 1. Solve the rational inequality \(\frac{7}{x-1} \leq 2\):

To solve this inequality:

1. **Subtract 2 from both sides** to get a single fraction:
   \[
   \frac{7}{x-1} - 2 \leq 0
   \]

2. **Express the inequality as a single fraction**:
   \[
   \frac{7 - 2(x-1)}{x-1} \leq 0
   \]
   Simplify the numerator:
   \[
   \frac{7 - 2x + 2}{x-1} \leq 0
   \]
   \[
   \frac{9 - 2x}{x-1} \leq 0
   \]

3. **Determine the critical points** by setting the numerator and denominator equal to zero:
   - Numerator: \(9 - 2x = 0\) ⟹ \(x = \frac{9}{2} = 4.5\)
   - Denominator: \(x - 1 = 0\) ⟹ \(x = 1\)

4. **Plot the critical points** on a number line and test the sign of the expression \(\frac{9 - 2x}{x-1}\) in the intervals divided by these points: \((-\infty, 1)\), \((1, 4.5)\), and \((4.5, \infty)\).

   - For \(x < 1\), the expression is positive.
   - For \(1 < x < 4.5\), the expression is negative.
   - For \(x > 4.5\), the expression is positive.

5. **Include the points where the inequality is less than or equal to zero**:
   - The solution is \(1 < x \leq 4.5\).

### 2. Solve for \(x\) in the rational inequality \(\frac{2x + 7}{x^2 - 2} \geq 0\):

1. **Determine the critical points** by setting the numerator and denominator equal to zero:
   - Numerator: \(2x + 7 = 0\) ⟹ \(x = -\frac{7}{2} = -3.5\)
   - Denominator: \(x^2 - 2 = 0\) ⟹ \(x = \pm\sqrt{2}\)

2. **Plot the critical points** on a number line: \(-\infty, -\sqrt{2}, -3.5, \sqrt{2}, \infty\).

3. **Test the sign** of the expression \(\frac{2x + 7}{x^2 - 2}\) in each interval:
   - For \(x < -\sqrt{2}\), the expression is negative.
   - For \(-\sqrt{2} < x < -3.5\), the expression is positive.
   - For \(-3.5 < x < \sqrt{2}\), the expression is negative.
   - For \(x > \sqrt{2}\), the expression is positive.

4. **Include the points where the inequality is greater than or equal to zero**:
   - The solution is \(x \in (-\sqrt{2}, -3.5] \cup [\sqrt{2}, \infty)\).

Would you like any more details or have any further questions?

Here are some related questions you might consider:
1. How can you check if your solution to a rational inequality is correct?
2. What happens to the inequality if you multiply or divide both sides by a negative number?
3. Can you solve a rational inequality by graphing? How?
4. What are some real-life applications of solving rational inequalities?
5. How would the solution change if the inequality signs were strict (\(<\) or \(>\)) instead of inclusive (\(\leq\) or \(\geq\))?

**Tip:** Always remember to check for any restrictions on the variable, especially where the denominator could be zero!

Learn how to solve rational inequalities step-by-step with detailed explanations and examples. This page covers solving rational inequalities involving rational functions and inequalities. Suitable for high school students and anyone interested in advanced algebraic problem-solving.

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