Lesson 97: Graphing Linear Inequalities

New Concept

A solution of a linear inequality is any ordered pair that makes the inequality true.

What’s next

Next, you’ll learn to visualize these solutions by graphing boundary lines and shading the correct half-plane on the coordinate grid.

Property

A solution of a linear inequality is any ordered pair that makes the inequality true.

Explanation

Think of it like a true/false test for coordinates! You plug the $(x, y)$ values from an ordered pair into the inequality. If the resulting statement is true, you've found a solution and that point is in the club! If it's false, that point is not invited to the party.

Examples

Is $(2, 5)$ a solution of $y > 2x + 1$? $5 > 2(2) + 1$ becomes $5 > 5$, which is false. Not a solution.
Is $(1, 8)$ a solution of $y \ge 3x + 5$? $8 \ge 3(1) + 5$ becomes $8 \ge 8$, which is true. It is a solution.
Is $(-3, 4)$ a solution of $y < -2x - 1$? $4 < -2(-3) - 1$ becomes $4 < 5$, which is true. It is a solution.

Property

The boundary line is a dashed line when the inequality contains the symbol < or >. The boundary line is a solid line when the inequality contains the symbol $\le$ or $\ge$.

Explanation

Imagine the boundary line is a fence for the solution zone. A solid line ($\le, \ge$) is a sturdy fence, meaning points on the fence are included in the solution area. A dashed line ($<, >$) is just for show; the points on that line are not part of the solution.

Examples

The graph of $y > 3x - 2$ has a dashed boundary line because points on the line are not solutions.
The graph of $y \le -x + 4$ has a solid boundary line because points on the line are included as solutions.

Property

To graph an inequality, graph the boundary line (solid or dashed). Then, use a test point, often $(0, 0)$, to decide which half-plane to shade. If the test point makes the inequality true, shade its entire region.

Explanation

First, draw your line—solid for 'or equal to,' dashed for not. Then, pick a test point not on the line, like $(0,0)$. If it makes the inequality true, shade its whole side! If it's false, you shade the other side. It’s all about finding the 'true' zone!

Examples

Graph $y < x + 2$. Draw a dashed line for $y = x + 2$. Test $(0, 0)$: $0 < 0 + 2$ is true. Shade the half-plane containing $(0, 0)$.
Graph $y \ge -2x + 1$. Draw a solid line for $y = -2x + 1$. Test $(0, 0)$: $0 \ge -2(0) + 1$ is false. Shade the half-plane that does not contain $(0, 0)$.

Let's turn an inequality into a map of infinite solutions on the coordinate plane. This example focuses on the key idea of graphing a linear inequality.

Example Problem

Graph the inequality $y \ge -\frac{1}{2}x + 1$.

Step-by-Step

1. First, graph the boundary line $y = -\frac{1}{2}x + 1$. Because the inequality symbol is $\ge$, we will use a solid line.
2. Next, we need to decide which side of the line to shade. We can use an ordered pair as a test point. The point $(0, 0)$ is a good choice since it is not on the line.
3. Substitute the coordinates of the test point into the inequality.

4. Simplify the expression.

5. This statement is false. The point $(0, 0)$ is not a solution. Therefore, we shade the half-plane that does not contain the point $(0, 0)$.