sketch the solution to each system of inequalities

x+2y less than or equal too -4
3x-2y> -4

To sketch the solution to the system of inequalities:

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

### Step-by-step guide:

1. **Convert inequalities to equations to find boundary lines:**

   - For \( x + 2y = -4 \), solve for \( y \):
     \[
     2y = -x - 4
     \]
     \[
     y = -\frac{1}{2}x - 2
     \]
     This is the equation of the boundary line for the first inequality.

   - For \( 3x - 2y = -4 \), solve for \( y \):
     \[
     -2y = -3x - 4
     \]
     \[
     y = \frac{3}{2}x + 2
     \]
     This is the equation of the boundary line for the second inequality.

2. **Graph the boundary lines:**
   
   - The line \( y = -\frac{1}{2}x - 2 \) (for the first inequality) will be solid because of the "less than or equal to" (≤) condition.
   - The line \( y = \frac{3}{2}x + 2 \) (for the second inequality) will be dashed because of the "greater than" (>) condition.

3. **Shading the regions:**
   
   - For \( x + 2y \leq -4 \), shade the region below or on the line \( y = -\frac{1}{2}x - 2 \).
   - For \( 3x - 2y > -4 \), shade the region above the line \( y = \frac{3}{2}x + 2 \).

4. **The solution region:**
   The solution to the system of inequalities is the region where the shaded areas overlap.

### Let's summarize the steps visually:
- Solid line for \( y = -\frac{1}{2}x - 2 \) (shade below it).
- Dashed line for \( y = \frac{3}{2}x + 2 \) (shade above it).
- The overlapping shaded region is the solution.

I can sketch this system for you if you'd like to see the visual representation. Let me know if you'd like to proceed with that!

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Do you want more details on the steps?  
Here are some related questions:
1. What is the importance of boundary lines in inequalities?
2. How do you identify which region to shade for an inequality?
3. Can a system of inequalities have no solution?
4. How do solid and dashed lines differ in graphing inequalities?
5. How do you test a point to verify the correct shaded region?

**Tip:** Always test a point (like the origin) to confirm which side of the boundary line to shade.

sketch the solution to each system of inequalities

x + 2y ≤ -4
3x - 2y > -4

Learn how to graph and sketch the solution to the system of inequalities x + 2y ≤ -4 and 3x - 2y > -4. This guide explains boundary lines, shading regions, and identifying the overlapping solution. Suitable for students in Grades 8-10.

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