How do you graphically represent the inequality x + y >= 5?

First, consider the corresponding equation $x + y = 5$. To draw this line, find two points that satisfy the equation.

• If $x = 0$, then $0 + y = 5 \implies y = 5$. So, the point is $(0, 5)$.
• If $y = 0$, then $x + 0 = 5 \implies x = 5$. So, the point is $(5, 0)$. Plot these two points and draw a straight line through them. Since the inequality is $x + y \ge 5$ (which includes "equal to"), the line should be continuous (solid), not dashed.

Choose a test point that is not on the line. The easiest point to use is often the origin $(0, 0)$.
Substitute $x = 0$ and $y = 0$ into the original inequality $x + y \ge 5$:

$$0 + 0 \ge 5$$ $$0 \ge 5$$

This statement is false.

Since the test point $(0, 0)$ does not satisfy the inequality, the region containing $(0, 0)$ is not part of the solution. Therefore, shade the region on the opposite side of the line from $(0, 0)$. In this case, $(0,0)$ is below and to the left of the line $x+y=5$, so the region above and to the right of the line should be shaded.

The graph in the image correctly shows the solid line $x+y=5$ and the region above and to the right of it shaded, representing all points $(x,y)$ for which $x+y \ge 5$.