The inequality given is $y \geq -x - 2$, which is a linear inequality. Let's break this down step by step for graphing:

Step 1: Graph the Boundary Line

• First, graph the line corresponding to the equality part of the inequality, $y = -x - 2$.
  • This is a straight line with slope $-1$ and y-intercept $-2$.
  • You plot the point $(0, -2)$ (where the line crosses the y-axis).
  • From that point, use the slope $-1$, which means for each step right (positive x-direction), go down by one unit. Alternatively, for each step left, go up by one unit.
  • This boundary line should be solid because the inequality is $\geq$, which means points on the line are included in the solution.

Step 2: Shade the Appropriate Region

• Since the inequality is $y \geq -x - 2$, you will shade above the line, because "greater than" refers to the y-values that are higher than the line.

Observations

In the image, only a small red section is graphed, and it looks like an incomplete or incorrect attempt to plot the inequality.

Would you like a more detailed walkthrough or help with the rest of the graph? Here are five related questions to deepen understanding:

1. What is the importance of using a solid or dashed line when graphing inequalities?
2. How do you determine the correct region to shade for inequalities?
3. What happens if the inequality was $y < -x - 2$?
4. Could you explain how slope affects the steepness and direction of a line?
5. What does it mean geometrically when we reverse the inequality symbol?

Tip: When plotting inequalities, always test a point (like $(0, 0)$) to verify if it satisfies the inequality and helps confirm the correct region to shade.