Solving Linear Inequalities with a Comparison to Linear Equations

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SWBAT solve and graph linear inequalities in one variable

Big Idea

Use visual representations to help students understand differences between solving linear inequalities and solving linear equations.

Warm Up

10 minutes

I intend for this Warm Up to take about 10 minutes for the students to complete and for me to review with the class.  The students perform different math operations to determine that multiplying or dividing by a negative number changes the meaning of the inequality.  In this work, multiplying or dividing both sides of the inequality by a negative number makes the inequality false, therefore the direction of the inequality sign must change.

As they complete this work, I expect that some of my students may not immediately recognize that -2 is greater than -3. If I come across this situation, I plan to have the students sketch a number line. This representation makes it easier for them to see that -2 is not less than -3. As they look at the graph I may need to ask probing questions to help them understand what they need to do to make the inequality statement true.

I have provided a Warm Up Key of what I expect students to complete on the Warm Up.

Guided Practice

15 minutes

After reviewing the Warm Up with my students, we will undertake some Guided Practice as a class. I plan to work the first two problems with my students as a class. I will ask students to attempt Problems 1 and 2, then we will:

  • check over the answers
  • review the meaning of the statements
  • discuss the shading of the solutions

I expect the students to take about 15 minutes to complete the Guided Practice and for us to review the answers as a class.  I model this in the video below.


To conclude the Guided Practice, my students complete a Venn Diagram using lists they have created naming differences and likenesses of solving inequalities and equations. To bring the activity to closure, we'll discuss the Venn Diagram. As we do I will encourage students to make changes as they increase their knowledge of solving linear inequalities.  I have placed a Venn Diagram-key in the resources of a possible completed Venn Diagram.

Partner Work

15 minutes

After the Guided Notes, I have students work on a Partner Worksheet of Inequalities that I found at the following website:

Each student should have a copy of the Partner Worksheet.

I instruct the students to attempt the first five problems on the partner worksheet.  One student works the five problems on the left column, and their table partner works the first five problems in the right column. Then the students check their work against their table partner.  The students have worked different problems, but the answers to number one should be the same, number two should be the same, etc., so that students may check their work.  If the students have different answers, I instruct them to look for their mistakes, correct the mistakes, and write a complete sentence about that correction on the paper.  By critiquing each others work (MP3), students are learning from each other's mistakes and corrections.

Depending on the time, I allow students to keep working and comparing the problems in each column.  With about 10 minutes remaining in class, I move to the Closure section of the lesson.  I assign the remainder of the worksheet as homework to be handed in the following day.  Each student should complete the column that they are working on as homework.  More advanced students may be able to switch columns and complete the other column as well.




10 minutes

During the last 10 minutes of today's class, I discuss the Closure of the lesson with the class.  I post the completed Venn Diagram from the Guided Practice section, and we compare it to the mistakes and corrections that the students made on the Partner Worksheet.  Some students may not have completed the Partner Worksheet yet, but I stop the lesson for all of the students to share what they have done up to this point.  

For example, in one of the differences of the Venn Diagram, it states that the inequality symbol changes direction when the inequality is multiplied or divided by a negative number on both sides. If a set of table partners made a mistake on the Partner worksheet to not change the direction of the symbol when they multiplied or divided by a negative, we discuss it.  If a student did not know how to graph the equation as one solution, we compare it to the Venn diagram. We discuss how an inequality has infinitely many solutions shown by shading, and an equation has one solution shown by one closed circle on that number.  Comparing the rules of solving linear inequalities with linear equations from the Venn diagram to specific examples from the Partner worksheet helps students make the connection between theory and practice.