# Lesson 24

Solutions to Systems of Linear Inequalities in Two Variables

### Lesson Narrative

Earlier in the unit, students solved systems of linear equations in two variables. They also found solutions to linear inequalities in two variables. In this lesson, students build on those understandings to find the solutions to systems of linear inequalities in two variables.

Students learn that two linear inequalities that represent the constraints in the same situation form a system of inequalities, and that the solutions to the system include all numbers that satisfy both constraints simultaneously. Graphically, the solution set of the system is represented by the region where the graphs of the individual inequalities overlap.

Throughout the lesson, students endeavor to find values that satisfy multiple constraints, both in terms of a situation and in absence of one. The process requires that students make sense of problems and persevere in solving them (MP1) and to reason quantitatively and abstractly (MP2).

### Learning Goals

Teacher Facing

• Given descriptions and graphs that represent two constraints, identify values that satisfy each constraint and those that satisfy both constraints simultaneously.
• Understand that the solution set of a system of inequalities in two variables is comprised of any pair of values that make both inequalities true, and that it is represented graphically by the region where the graphs overlap.
• Write systems of inequalities in two variables, use technology to graph the solutions, and interpret the solutions in context.

### Student Facing

• Let’s look at situations where two constraints (that can be expressed by inequalities) must be met simultaneously.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (If students typically access the digital version of the materials, Desmos is available under Math Tools.)

Be prepared to display an embedded applet for all to see.

### Student Facing

• I can write a system of inequalities to describe a situation, find the solution by graphing, and interpret points in the solution.
• I know what is meant by "the solutions to a system of inequalities" and can describe the graphs that represent the solutions.
• When given descriptions and graphs that represent two different constraints, I can find values that satisfy each constraint individually, and values that satisfy both constraints at once.

Building On

Building Towards

### Glossary Entries

• solutions to a system of inequalities

All pairs of values that make the inequalities in a system true are solutions to the system. The solutions to a system of inequalities can be represented by the points in the region where the graphs of the two inequalities overlap.

• system of inequalities

Two or more inequalities that represent the constraints in the same situation form a system of inequalities.