Lesson 39: Graphing Linear Inequalities in Two Variables

New Concept

A linear inequality in two variables relates two variables, often $x$ and $y$, with an inequality sign.

What’s next

Next, you’ll learn to visualize these relationships by graphing them on a coordinate plane, separating solutions from non-solutions with a boundary line.

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.

To graph a linear inequality, first graph the related linear equation. Make the line dashed for $<$ or $>$ and solid for $\leq$ or $\geq$. This is the boundary line, which separates the plane into two half-planes.

For the inequality $y < 2x + 5$, the boundary line is $y = 2x + 5$, and it must be dashed.
For the inequality $4x + 3y \geq 12$, the boundary line is $4x + 3y = 12$, and it must be solid.

The boundary line is like a fence in a massive field. A solid line is a solid fence, meaning points on it are part of the solution (like for $\leq$). A dashed line is like an electric fence you can't touch, so points on it are not included (like for $<$). It perfectly divides the graph into two zones.

To determine which half-plane contains the solutions, use a test point not on the boundary line. If the test point satisfies the inequality, shade that half-plane. The point $(0, 0)$ is a good test point when available.

For $3y + x \geq -9$, test $(0, 0)$: $3(0) + 0 \geq -9 \rightarrow 0 \geq -9$. This is true, so shade the half-plane containing $(0, 0)$.
For $y > 4x + 2$, test $(0, 0)$: $0 > 4(0) + 2 \rightarrow 0 > 2$. This is false, so shade the half-plane that does not contain $(0, 0)$.

Think of the test point as a secret agent. You send the easiest agent, $(0, 0)$, into one of the half-planes. If it reports back 'True,' that whole region is the solution, and you shade it in! If it reports 'False,' you know the solutions are all hiding in the other half-plane, so you shade that one instead.

When an inequality is in slope-intercept form ($y < mx + b$), you can shade by looking at the inequality sign. Shade above the line for $>$ and $\geq$, and shade below the line for $<$ and $\leq$.

For $y < -\frac{1}{5}x + 4$, the symbol is $<$, so you shade below the dashed boundary line.
For $-2y < -10x - 2$, first solve for $y$: $y > 5x + 1$. The symbol is $>$, so shade above the dashed boundary line.

This is a super-fast shortcut! Once your inequality is solved for $y$, the sign tells you everything. 'Greater than' ($>$ or $\geq$) means you want the bigger $y$-values, so you shade 'up' or above the line. 'Less than' ($<$ or $\leq$) means you want the smaller $y$-values, so you shade 'down' or below it.