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!

---

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.