https://file.aiquickdraw.com/mathbot/file/1725999149397ewmf2tzr.jpg

Let's go through the even-numbered inequalities one by one and solve them for graphing:

### 34. \( x - 2y \leq -6 \)

1. Start by solving for \( y \):
   \[
   -2y \leq -x - 6
   \]
   \[
   y \geq \frac{1}{2}x + 3
   \]
   (Note: The inequality flips when dividing by a negative number.)

2. To graph this inequality:
   - The boundary line is \( y = \frac{1}{2}x + 3 \) (a solid line since it is \( \leq \)).
   - The slope is \( \frac{1}{2} \), and the y-intercept is 3.
   - Shade above the line because the inequality is \( \geq \).

---

### 36. \( x = -2y + 10 \)

1. Solve for \( y \):
   \[
   2y = -x + 10
   \]
   \[
   y = -\frac{1}{2}x + 5
   \]

2. Graph this:
   - The boundary line is \( y = -\frac{1}{2}x + 5 \) (a solid line since there is no inequality symbol).
   - The slope is \( -\frac{1}{2} \), and the y-intercept is 5.
   - This is just a line without shading since there’s no inequality.

---

### 38. \( y + 1 < -2(x - 3) \)

1. Distribute on the right side:
   \[
   y + 1 < -2x + 6
   \]
2. Subtract 1 from both sides:
   \[
   y < -2x + 5
   \]

3. Graph this:
   - The boundary line is \( y = -2x + 5 \) (a dashed line because it’s \( < \)).
   - The slope is \( -2 \), and the y-intercept is 5.
   - Shade below the line because the inequality is \( < \).

---

Would you like to explore any specific graphs or need help with any specific steps?

Here are 5 related questions to deepen understanding:
1. What happens when an inequality flips during solving?
2. How do you graph an inequality with a horizontal or vertical line?
3. Why do we shade above or below the line depending on the inequality?
4. How do dashed and solid lines represent different inequalities in graphing?
5. What is the significance of the slope in linear inequalities?

**Tip**: Always check whether the inequality is strict (\(<\) or \(>\)) or non-strict (\(\leq\) or \(\geq\)) to determine whether to use a dashed or solid line when graphing.

Solve all even-numbered inequalities from the given exercise set and sketch the graph.

Learn how to solve and graph linear inequalities by going through even-numbered problems step-by-step. This problem set involves key algebraic concepts like slope-intercept form and properties of inequalities, suitable for students in Grades 9-12.

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