5-5 Graphing Inequalities in Two Variables

Property

A linear inequality in two variables relates $x$ and $y$ using an inequality symbol. A solution is any coordinate pair $(x, y)$ that makes the mathematical statement true.

To graph these infinite solutions, you must first draw a boundary line. You find this line by temporarily replacing the inequality symbol with an equal sign to get a standard linear equation (e.g., $y = mx + b$). This boundary line slices the entire coordinate grid into two distinct regions called half-planes. The solution to the inequality will be all the points located in exactly one of these half-planes.

Examples

• Checking a Solution: Is $(2, 9)$ a solution for $y > -3x + 5$?

Substitute $x=2$ and $y=9$: $9 > -3(2) + 5 \rightarrow 9 > -1$. This is true, so $(2, 9)$ is a solution.

• Setting up the Boundary Line (Slope-Intercept): Graph $2x + 4y \leq 8$.

Rewrite as $2x + 4y = 8$. Solve for $y$: $4y = -2x + 8 \rightarrow y = -0.5x + 2$. The boundary line has a slope of -0.5 and a y-intercept of 2.

• Setting up the Boundary Line (Intercepts): Graph $3x - 5y > 15$.

Rewrite as $3x - 5y = 15$.
Set $x=0$ to find the y-intercept: $-5y = 15 \rightarrow y = -3$. Point: $(0, -3)$.
Set $y=0$ to find the x-intercept: $3x = 15 \rightarrow x = 5$. Point: $(5, 0)$. Plot these two points to draw the boundary.

Property

When drawing the boundary line of a linear inequality, you must use specific formatting to show whether the line itself is part of the solution set:

• Dashed Line: Use for strictly "less than" ($<$) or "greater than" ($>$). Points sitting exactly on a dashed line are NOT solutions.

• Solid Line: Use for "less than or equal to" ($\leq$) or "greater than or equal to" ($\geq$). Points sitting exactly on a solid line ARE solutions.

Property

To find which half-plane contains the solutions (and should be shaded), use the Test Point Method:

1. Graph the boundary line (solid or dashed).
2. Choose a simple test point that is strictly NOT on the boundary line. The origin $(0, 0)$ is always the best choice, unless the line passes directly through it.
3. Substitute the coordinates of the test point into the original inequality.
4. If the result is a TRUE statement, shade the entire half-plane that contains the test point. If it is FALSE, shade the opposite half-plane.

Examples

• Using $(0,0)$ as a test point: Graph $x - 3y < 6$.

Draw the dashed line $x - 3y = 6$. Test the origin $(0,0)$: $0 - 3(0) < 6$ simplifies to $0 < 6$. This is a TRUE statement. Therefore, shade the side of the line that includes the point $(0,0)$.

• When $(0,0)$ is on the line: Graph $y \leq 2x$.

Draw the solid line $y = 2x$. Since this line passes exactly through the origin, we must choose a different point, like $(3, 1)$. Test it: $1 \leq 2(3)$ simplifies to $1 \leq 6$. This is TRUE. Shade the side containing the point $(3, 1)$.

• A False Result: Graph $y > 4$.

Draw a dashed horizontal line at $y = 4$. Test $(0,0)$: $0 > 4$ is FALSE. Shade the side that does NOT contain $(0,0)$, which is the region above the line.

Explanation

Because the boundary line cuts the graph perfectly in half, all the correct answers live together on one side, and all the wrong answers live together on the other. This means you don't have to test a hundred different points! You only need to test one single point to scout out the territory. If your scout point tells the truth, its entire side is the winner. If it lies, the other side wins.