Lesson 4: Linear Inequalities in Two Variables

Property

A linear inequality is an inequality that can be written in one of the following forms:
$A x + B y > C$
$A x + B y \geq C$
$A x + B y < C$
$A x + B y \leq C$
where $A$ and $B$ are not both zero.

Examples

• The statement $3x - y < 7$ is a linear inequality.
• The statement $y \geq 2x + 5$ is a linear inequality.
• The statement $x < 4$ is a linear inequality where the coefficient of $y$ is $0$.

Explanation

A linear inequality describes a relationship that isn't strictly equal. Instead of a single line of solutions, it represents a whole region on the coordinate plane. Think of it as a 'more than' or 'less than' zone with infinite solutions.

Property

The line with equation $Ax + By = C$ is the boundary line that separates the region where $Ax + By > C$ from the region where $Ax + By < C$.

For $Ax + By < C$ or $Ax + By > C$, the boundary line is not included in the solution, and the line is dashed.

For $Ax + By \leq C$ or $Ax + By \geq C$, the boundary line is included in the solution, and the line is solid.

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.