Solving a System of Inequalities

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SWBAT solve a system of linear inequalities.

Big Idea

Solve a system of inequalities by testing a point and by interpreting solutions from slope-intercept form.

Warm Up

10 minutes

The objective of this lesson is to solve a system of inequalities, so I begin by accessing the students' prior knowledge of solving a linear equation and a linear inequality in one variable. I allow the students 5 minutes to complete the Warm-Up.  I review the warm up with the class to introduce the lesson.  

I discuss today's warm-up in the video below:

After the warmup, the main purpose of today's lesson is to build on known concepts by making connections. First, to solving a linear inequality in two variables and then to solving a system of linear inequalities in two variables. I discuss with the students how inequalities place restrictions on the problem, but there are many solutions.  These restrictions I refer to as boundaries or constraints.  

Guided Notes

10 minutes

In the Guided Notes I discuss:

  • The meaning of a solution to a system
  • How to solve and graph an inequality
  • The test point method compared to the slope-intercept method

I set up the notes for students. My plan is for them to develop an understanding of how the symbols are related to the solutions of the inequality. We will also review the meaning of this relationship. After completing the Guided Notes, students should recognize that solutions are contained in the intersection of the shaded regions, and, on the solid boundary lines in some cases.  

Students are also required to be able to apply the following skills:

  • Solve for y when rewriting inequalities in slope intercept form.
  • Change the direction of the inequality symbol when multiplying or dividing each side of the inequality by a negative number
  • Graphing a line from a given equation
  • Substituting values for x and y into an equation to determine if the inequality is true or false
  • Graphing the solution to an inequality and a system of inequalities

Throughout the warm up, guided notes, practice, and lesson, students should gain a conceptual understanding of why these different methods for solving a system of equations work.  I ask the following questions throughout the lesson.

  1. When an inequality is false on one side, why can we determine that the other side is the solution?
  2. What form do the inequalities need to be in to apply the slope-intercept method?
  3. What form do the inequalities need to be in to apply the test point method?
  4. Can a point on a boundary line be used? Why or Why not?


Independent Practice

15 minutes

While students are working the Independent Practice, they are allowed to talk with their assigned elbow partner for math questions, but they must stay on task.  I walk around the room while students are working to monitor progress and to assist students one on one or in pairs as needed. I use these questions continue to guide students forward in their productive struggle.

  1. Do you have to graph a linear inequality in one variable on a number line or a coordinate plane?
  2. If you graph a linear inequality in one variable on a coordinate plane, what type of line is graphed?
  3. On problem 10 on the practice, what is the difference between solving a system of equations that is parallel compared to solving a system of inequalities that are parallel?
  4. What do you think is the difference between solving a system of inequalities with the same line compared to a system of inequalities that are the same line?
  5. Is it possible to graph number 7 on a number line, or must you graph it on a coordinate plane?  Why or Why not?
  6. What is the difference in the solutions to a system of equations with different slopes that are intersecting compared to a system of linear inequalities that are intersecting?

In the last problem, students have not been introduced to graphing quadratic functions this year. Some students do remember graphing quadratics previously in 8th grade.  This provides a good teaching moment to discuss a strategy to use when graphing an unfamiliar function.  A t-table is a strategy to use to substitute random numbers for x and solve for y to determine the points of the graph and the shape of the function.


Check Work with Online Activity

10 minutes

In closing, we discuss how a linear inequality in one variable can be graphed on the number line or on a coordinate plane.

I also have students check their work by entering into an online graphing tool.  I instruct students that the equations or relations should be entered in standard form Ax + By = C.  If the inequality has only one variable, a 0 must be entered as the coefficient for the other variable.