Let’s solve this step-by-step to determine which graph corresponds to the inequality $y \geq -2x - 2$.

Step 1: Graph the boundary line $y = -2x - 2$

• Slope = $-2$ (meaning the line decreases 2 units vertically for every 1 unit it moves horizontally).
• Y-intercept = $-2$, so the line crosses the y-axis at $(0, -2)$.

Step 2: Plot key points

• When $x = 0$, $y = -2(0) - 2 = -2$ → Point: $(0, -2)$.
• When $x = -2$, $y = -2(-2) - 2 = 4 - 2 = 2$ → Point: $(-2, 2)$.

These two points, $(0, -2)$ and $(-2, 2)$, lie on the boundary line.

Step 3: Shading the appropriate region

• Since the inequality is $y \geq -2x - 2$, we shade the region above or on the line, where $y$ is greater than or equal to the line’s value.

Step 4: Identify the correct graph

• Look for a line passing through $(-2, 2)$ and $(0, -2)$ with shading above the line.

Matching the graphs:

• Graph D matches the correct boundary line and correctly shades the region above it.

Final Answer:

The correct answer is B. Graph D.

Do you want more details on how graph inequalities work or have any questions?

Here are 5 related questions you might find helpful:

1. How do you determine when to use a solid vs. dashed line in inequalities?
2. What does the slope represent in a linear function or inequality?
3. How do you determine which side of the line to shade for inequalities?
4. Can you explain how to solve systems of linear inequalities graphically?
5. What are other ways to verify if a point lies in the solution set of an inequality?

Tip: When graphing inequalities, always test a point (like $(0,0)$) to confirm which region satisfies the inequality.