Lesson 5: Linear Inequalities in Two Variables

New Concept

Explore linear inequalities in two variables, like $ax + by \leq c$. You'll learn to graph their solutions as half-planes and find the solution regions for systems of inequalities, which can represent real-world constraints.

What’s next

Next, you'll walk through graphing methods with interactive examples. Then, you'll apply these skills to solve systems of inequalities through a series of practice cards.

Property

A linear inequality can be written in the form

Examples

• To graph $3x - y > 6$, we first solve for $y$. This gives $-y > -3x + 6$, which becomes $y < 3x - 6$ after dividing by $-1$. We draw a dashed line for $y = 3x - 6$ and shade the half-plane below it.

• To graph $x + 2y \leq 8$, we solve for $y$ to get $2y \leq -x + 8$, or $y \leq -\frac{1}{2}x + 4$. We draw a solid line for $y = -\frac{1}{2}x + 4$ and shade the region below it.

Property

To graph an inequality using a test point:

1. Graph the boundary line.
2. Choose a test point not on the line, like $(0, 0)$ if possible.
3. Substitute the point's coordinates into the inequality.

a. If the statement is true, shade the half-plane containing the test point.
b. If the statement is false, shade the other half-plane.

Examples

• To graph $4x + 3y > 12$, we first graph the line $4x+3y=12$. Let's test the origin $(0,0)$: $4(0) + 3(0) > 12$, or $0 > 12$, is false. Therefore, we shade the side of the line that does not contain the origin.

• To graph $y \leq 2x$, we graph the line $y=2x$. Since this line passes through $(0,0)$, we must choose a different test point, like $(3,1)$. Is $1 \leq 2(3)$? Yes, $1 \leq 6$ is true. We shade the side containing $(3,1)$.

Property

An inequality like $x \geq k$ represents $x + 0y \geq k$. Its graph is a vertical line at $x=k$ and a shaded half-plane. Similarly, an inequality like $y < k$ has a horizontal boundary line at $y=k$ and a shaded half-plane.

Examples

• To graph $x < 4$, draw a dashed vertical line at $x=4$. Since the x-coordinates must be less than 4, shade the entire half-plane to the left of the line.

• To graph $y \geq -2$, draw a solid horizontal line at $y=-2$. Since the y-coordinates must be greater than or equal to -2, shade the entire half-plane above the line.

Property

The solution to a system of inequalities is the set of all points that satisfy every inequality in the system.

The graph of the system is the intersection, or overlapping area, of the shaded regions for each individual inequality.

The corner points of this region are called vertices.