Lesson 109: Graphing Systems of Linear Inequalities

New Concept

A system of linear inequalities is a set of linear inequalities with the same variables.

What’s next

Next, you’ll translate these algebraic constraints into visual solution sets on the coordinate plane, mastering the boundaries of what is possible.

Property

A system of linear inequalities is a set of linear inequalities with the same variables.

Explanation

It's a team game! A point is only a 'solution' if it makes every single inequality on the team happy. Graphically, this is the region where all the shaded areas from each inequality overlap.

Examples

For the system $y \le x+1$ and $y > -2$, the point $(1,1)$ is a solution since $1 \le 2$ and $1 > -2$ are both true.

Property

Graph each inequality on the same coordinate plane. The solution to the system is the region where the shaded areas of all inequalities overlap.

Explanation

It's like layering transparent colored sheets. The area where colors mix is your solution! Use a dashed line for $<$ or $>$ and a solid line for $\le$ or $\ge$.

Examples

For the system $y > x$ and $y \le 4$, the solution is the area where the shading for both graphs overlaps.

Each inequality splits the plane in two; let's find the single region where they both agree. This example illustrates the core process of solving systems by graphing.

Example Problem

Graph the system: $y > \frac{1}{2}x - 2$ and $y \le -2x + 5$.

Property

Parallel boundary lines can result in no solution, a solution between the lines, or one solution being a subset of another.

Explanation

Parallel lines can be tricky! If their shaded areas point away from each other, there is no overlap and thus no solution. If they point toward each other, the solution is the strip of land between them.

Examples

$y \ge x+3$ and $y \le x-1$ have no solution as the shaded regions never intersect.