Q. Which is the graph of the linear inequality \(y \ge -x – 3\)?

Problem: Graph the linear inequality \( y \ge -x – 3 \)

Step 1 – Identify the boundary line

The boundary line is the equation

\[
y = -x – 3
\]

Because the inequality is \(\ge\), draw this boundary as a solid line (points on the line are included in the solution set).

Step 2 – Find the intercepts of the boundary line

To find the y-intercept, set \( x = 0 \):

\[
y = -(0) – 3 = -3
\]

So the y-intercept is \((0,-3)\).

To find the x-intercept, set \( y = 0 \):

\[
0 = -x – 3
\]
\[
x = -3
\]

So the x-intercept is \((-3,0)\).

Step 3 – Determine the shading region using a test point

Pick a test point not on the line, for example \((0,0)\). Substitute it into the inequality:

\[
0 \ge -(0) – 3
\]
\[
0 \ge -3
\]

This is true, so the half-plane containing the origin should be shaded.

Step 4 – Describe the final graph

Draw the solid line \( y = -x – 3 \) through the points \((0,-3)\) and \((-3,0)\). Shade the region above the line (the half-plane containing the origin) to represent all solutions of the inequality.

Answer

\[
y \ge -x – 3
\]

The graph is the solid line \( y = -x – 3 \) with the region above the line shaded.