https://file.aiquickdraw.com/mathbot/file/1728388727207iga92y2i.jpg

The image contains two inequalities that need to be graphed:

1. \( x + 2y > 4 \)
2. \( y - 2x \leq 3 \)

### Let's solve each inequality step by step:

#### 1. Inequality: \( x + 2y > 4 \)
We need to express \( y \) in terms of \( x \):
\[
x + 2y > 4
\]
Subtract \( x \) from both sides:
\[
2y > -x + 4
\]
Now divide by 2:
\[
y > -\frac{1}{2}x + 2
\]
This represents the line \( y = -\frac{1}{2}x + 2 \), but because it’s a strict inequality (\( > \)), you will draw a dashed line for the boundary and shade the region above the line.

#### 2. Inequality: \( y - 2x \leq 3 \)
We also need to solve for \( y \):
\[
y - 2x \leq 3
\]
Add \( 2x \) to both sides:
\[
y \leq 2x + 3
\]
This represents the line \( y = 2x + 3 \), but since it’s a “less than or equal to” inequality (\( \leq \)), you will draw a solid line for the boundary and shade the region below the line.

### Graphing the inequalities:

- For the inequality \( y > -\frac{1}{2}x + 2 \), plot the line \( y = -\frac{1}{2}x + 2 \) using points such as \( (0, 2) \) and \( (4, 0) \). Since it's a "greater than" inequality, shade above the dashed line.
  
- For the inequality \( y \leq 2x + 3 \), plot the line \( y = 2x + 3 \) using points like \( (0, 3) \) and \( (-1, 1) \). Shade below the solid line.

I can generate the graph for you if needed! Would you like me to generate the graph? Let me know if you want any details or further explanation.

### Relative Questions:
1. What are the differences between solid and dashed lines in graphing inequalities?
2. How do you determine which region to shade when graphing inequalities?
3. How would the graph change if \( x + 2y \geq 4 \) instead of \( x + 2y > 4 \)?
4. What are some real-life situations where you would graph inequalities like these?
5. Can you explain how to find the intersection region of two inequalities?

### Tip:
When graphing inequalities, always check a test point (like the origin \( (0,0) \)) to ensure you are shading the correct region.

Graph the given inequalities:
1) x + 2y > 4
2) y - 2x ≤ 3

Learn how to graph the inequalities x + 2y > 4 and y - 2x ≤ 3. This step-by-step guide explains how to rearrange equations, plot linear boundaries, and determine shaded regions for algebra students in Grades 9-11.

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