1. **State the problem:** We need to graph the inequality $3x + 4y \geq 12$ on the Cartesian plane. 2. **Rewrite the inequality as an equation to find the boundary line:** $$3x + 4y = 12$$ This line divides the plane into two regions. 3. **Find the intercepts of the boundary line:** - For the $x$-intercept, set $y=0$: $$3x + 4(0) = 12 \implies 3x = 12 \implies x = 4$$ - For the $y$-intercept, set $x=0$: $$3(0) + 4y = 12 \implies 4y = 12 \implies y = 3$$ 4. **Plot the boundary line:** The line passes through points $(4,0)$ and $(0,3)$. 5. **Determine which side of the line to shade:** Choose a test point not on the line, for example $(0,0)$. Substitute into the inequality: $$3(0) + 4(0) = 0 \geq 12$$ This is false, so the region containing $(0,0)$ is NOT part of the solution. 6. **Shade the opposite side:** Shade the region on the side of the line opposite to $(0,0)$, which satisfies $3x + 4y \geq 12$. 7. **Include the boundary line:** Since the inequality is $\geq$, the boundary line itself is included (solid line). Final answer: The graph is the line through $(4,0)$ and $(0,3)$ with the region above (or to the side opposite $(0,0)$) shaded, including the line itself.