https://file.aiquickdraw.com/mathbot/file/1729642947051ueks8cc0.jpg

The system of inequalities in the image is:

1. \( 2x + 4y \leq 4 \)
2. \( 2x + y \leq 10 \)

To graph the solution set, follow these steps:

### 1. Rewrite each inequality in slope-intercept form (\( y = mx + b \)).

#### First inequality: \( 2x + 4y \leq 4 \)

- Solve for \( y \):
  \[
  4y \leq -2x + 4
  \]
  \[
  y \leq -\frac{1}{2}x + 1
  \]

#### Second inequality: \( 2x + y \leq 10 \)

- Solve for \( y \):
  \[
  y \leq -2x + 10
  \]

### 2. Graph the boundary lines.

- The line \( y = -\frac{1}{2}x + 1 \) has a slope of \( -\frac{1}{2} \) and a y-intercept at \( (0, 1) \).
- The line \( y = -2x + 10 \) has a slope of \( -2 \) and a y-intercept at \( (0, 10) \).

For both inequalities, the shaded region will be below the boundary lines because the inequalities are "less than or equal to" ( \( \leq \)).

### 3. Find the intersection points of the boundary lines.
Solve the system:
\[
y = -\frac{1}{2}x + 1
\]
\[
y = -2x + 10
\]
Substitute \( y = -\frac{1}{2}x + 1 \) into \( y = -2x + 10 \):
\[
-\frac{1}{2}x + 1 = -2x + 10
\]
Multiply through by 2 to eliminate fractions:
\[
-x + 2 = -4x + 20
\]
Simplify and solve for \( x \):
\[
3x = 18
\]
\[
x = 6
\]
Substitute \( x = 6 \) into \( y = -\frac{1}{2}x + 1 \):
\[
y = -\frac{1}{2}(6) + 1 = -3 + 1 = -2
\]
So, the intersection point is \( (6, -2) \).

### 4. Shade the appropriate region.

- Shade the region below both lines.
- The solution is the region where the shaded areas overlap.

Would you like me to graph this for you or provide further details?

Here are some related questions:
1. How do you solve systems of inequalities algebraically?
2. How does the slope of a line affect the inequality solution set?
3. What happens when inequalities are strict (using \( < \) or \( > \)) instead of \( \leq \)?
4. How can you verify the intersection point in a system of inequalities graphically?
5. How do you determine whether to shade above or below the boundary line?

**Tip:** Always plot the boundary lines first, and then use a test point (like \( (0,0) \) if not on the line) to check which side of the boundary to shade.

Graph the solution set of the following system of inequalities: 2x + 4y ≤ 4, 2x + y ≤ 10

This guide helps students learn how to graph the solution set of the system of inequalities: 2x + 4y ≤ 4 and 2x + y ≤ 10. It includes rewriting equations into slope-intercept form and finding intersection points to determine the overlapping shaded region. Suitable for students in Grades 9-11.

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