Solving Systems of Linear Inequalities

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When algebra 2 students dive into linear systems, they first learn how to solve systems of linear equations and then proceed with solving systems of linear inequalities, including graphing these systems.

These lessons may not be smooth sailing for all students, especially for students who lack the foundational knowledge of solving linear inequalities and are now asked to jump into solving systems of linear inequalities.

This is why it’s important that math teachers and homeschooling parents have the appropriate teaching strategies in place. To help out, we’ve created a list of strategies to teach this topic. Use these strategies in your classroom and see your students’ math knowledge soar!

How to Teach Solving Systems of Linear Inequalities

What Are Linear Inequalities?

You may want to start your lesson by reviewing what linear inequalities are. Remind students that a linear inequality is similar to a linear equation; however, instead of an equality sign, the linear inequality has an inequality sign (such as <, >, ≤, ≥).

We can also say that the expressions of the forms: ax < b, ax + b ≥ c, ax + by > c, ax + by ≤ c are called linear inequalities. Explain that you’ll work on two-variable linear inequalities. You can write a few examples of linear inequalities in two variables on the whiteboard, such as:

• 5x – y > – 2
• 2x + 3y > 1
• y ≥ 2x – 3
• y ≤ 3/2x + 3

What Is a System of Linear Inequalities?

After having reviewed linear inequalities, you can explain that a system of linear inequalities is simply a set of linear inequalities containing the same variables. Point out that this set consists of at least two linear inequalities and that we deal with these inequalities all at once.

Write a few examples on the whiteboard of systems of linear inequalities:

1. 3x + 7y ≥ 21

x – y ≤ 2

2. 2x + 3y ≤ 18

2x + y ≤ 10

– 2x + y ≤ 2

What Is Solving Systems of Linear Inequalities?

First, remind students that the solution to a two-variable linear inequality with x and y is an ordered pair (a, b) if the inequality remains true when we replace a and b for x and y.

You can also create a brief bell-work activity where students check if a given ordered pair is a solution to a certain linear inequality. For example, ask students to check which of these two ordered pairs (3, 4) or (1, 1) is a solution to the inequality x + 2y < 9.

Remind students that the graph of an inequality is the collection of all solutions to the inequality. Feel free to use this video by Khan Academy which provides step-by-step instructions on how to solve and graph linear inequalities in two variables, including worked out examples. You may also want to check out our article on solving inequalities in one variable.

Explain that the solution to a system of linear inequalities is an ordered pair that satisfies both inequalities, that is, an ordered pair that is a solution to all the inequalities in the system. Write a system of inequalities and ask students to check whether a specific ordered pair is a solution.

Point out that when solving systems of linear inequalities, we graph each inequality from the system on the same coordinate plane. The region where the graphs of all inequalities overlap represents the solution to the given system of linear inequalities.

Any ordered pair from this region will satisfy all inequalities in the system. You can use this video as a video lesson to help you illustrate the process of graphing each linear inequality and shading the region where the two inequalities in the system overlap in the coordinate plane.

Steps for Solving Systems of Linear Inequalities:

1. First, we’ll take a look at the first inequality in the system. We’ll solve the inequality for y, that is, we’ll isolate y.
2. We’ll then graph this inequality on the coordinate plane and shade the half-plane that satisfies the inequality. Point out that depending on the inequality sign, we graph the line as a solid or a dashed line.
• When the inequality sign in the linear inequality is ≥ or ≤, we’ll draw a solid line on the coordinate plane.
• When the inequality sign in the linear inequality is > or <, we’ll draw a dashed line in the coordinate plane.
1. Now we can shade the half-plane that satisfies the inequality.
2. Proceed to the second linear inequality in the system and repeat the three above steps.
3. Shade the region where all the half-planes intersect in a darker color. In cases where there is no such intersection region, we can infer that there is no solution to the given system of linear inequalities.

Activities to Practice Solving Systems of Linear Inequalities

Online Activity

This is a simple online activity that you can use at the end of your lesson on solving systems of linear inequalities. The only thing you’ll need to implement this activity in your classroom is a sufficient number of technical devices for each student.

This is an individual activity, which makes it suitable for homeschooling parents as well. Explain to students that they will solve several questions on their device, related to solving systems of linear inequalities.

More specifically, students need to identify which of the multiple answers represents the graph of the given system of linear inequalities. If they get stuck, they can also choose to get a hint or watch a video to help them out. End the activity with a brief discussion and reflection.

Pair Work
This activity will help students practice solving systems of linear inequalities. Use this Assignment Worksheet (Members Only) to implement this activity in your classroom. The worksheet contains diverse exercises on systems of linear inequalities.

Print out the above worksheet and make sure there is one copy per student. Divide students into pairs and hand out the printouts. Explain to students that they work individually to solve the exercises.

After a few minutes, they exchange their worksheets with the student in their pair and review each other’s work. They provide feedback to each other and identify any mistakes. By doing so, the students engage in peer tutoring.

Before You Leave…

If you enjoyed the teaching strategies on solving systems of linear inequalities that we shared in this article, you’ll want to check out our lesson that goes in-depth in teaching this topic!

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