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Let's work through this system of inequalities step-by-step to find and graph the feasible region.

### The system of inequalities is:

1. \( 2y + x \geq -2 \)
2. \( y + 3x \leq 9 \)
3. \( y \leq 0 \)
4. \( x \geq 0 \)

### Step-by-step solution:

1. **First inequality: \( 2y + x \geq -2 \)**

   Rewrite this inequality to express \( y \) in terms of \( x \):
   \[
   2y \geq -2 - x \quad \Rightarrow \quad y \geq -1 - \frac{x}{2}
   \]
   This is the boundary line \( y = -1 - \frac{x}{2} \), but since it’s \( \geq \), we will shade the region **above** the line.

2. **Second inequality: \( y + 3x \leq 9 \)**

   Solve for \( y \) in terms of \( x \):
   \[
   y \leq 9 - 3x
   \]
   This is the boundary line \( y = 9 - 3x \), but since it’s \( \leq \), we will shade the region **below** the line.

3. **Third inequality: \( y \leq 0 \)**

   This represents all points below or on the \( x \)-axis (since \( y = 0 \) is the boundary line).

4. **Fourth inequality: \( x \geq 0 \)**

   This represents all points to the right of or on the \( y \)-axis (since \( x = 0 \) is the boundary line).

### Finding the feasible region:

Now, we need to find where all these shaded regions overlap. We’ll plot the boundary lines for each inequality and determine the common area where all conditions are satisfied.

Would you like me to provide a visual representation of this solution or assist further with any clarifications?

Here are 5 follow-up questions that may help:

1. Do you understand how to rearrange inequalities to graph them?
2. Would you like more explanation on how to graph each inequality?
3. Should we calculate the points of intersection between the lines?
4. Do you need clarification on how to shade the regions?
5. Would you like to go over how to check points within the feasible region?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Graph the feasible region for the system of inequalities: 2y + x >= -2, y + 3x <= 9, y <= 0, x >= 0.

Learn how to graph the feasible region for the system of inequalities: 2y + x >= -2, y + 3x <= 9, y <= 0, and x >= 0. This problem covers key algebra concepts such as graphing linear inequalities and determining the feasible region where all inequalities intersect.

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