Absolute Value Inequalities

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Students will be to solve absolute value equations and inequalities graphically and algebraically.

Big Idea

Absolute value inequalities are used when modeling requires a range of solutions.

Warm Up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Absolute Value Inequalities which asks students to create their own questions from a scenario.

I also use this time to correct and record the previous day's Homework.  

Finding Absolute Value Equations and Inequalities Graphically

20 minutes

In the previous lesson, students solved absolute value equations.  Today, the students start by graphing these equations to find the solutions.  For detailed presentation notes please see the PowerPoint.  This is my students’ first opportunity to look at the graph of absolute value.  Once they have looked at the graphical solutions to a variety of absolute value equations with and without extraneous solutions, we will move on to inequalities. 

I have found that many students learn the procedure of solving absolute value inequalities without really understanding WHY they work.  Therefore, I chose to first introduce solve these eqautions graphically to help the students build a conceptual understanding (Math Practice 5).  I have the students write the solutions using inequalities, interval notation and a number line.  These multiple representations will help them better build a solid understanding.  We start with several greater than inequalities.  The students graph each using graphing technology and then write a generalized conclusion about this type of inequality.  I also have them draw a sketch showing a sample of the graphical solution to remind themselves.   They will then do the same thing with less-than inequalities.  Finally, they will look at two problems that have either no solution or infinite solution.  Again, specific procedures and scaffold is located in the PowerPoint.

Modeling and Solving Absolute Value Inequalities

18 minutes

Once the students have a conceptual understanding graphically, we move onto solving absolute value inequalities algebraically.  My goal is that they use their knowledge of the graphical representation to know whether they are dealing with a disjunction or a conjunction. This transfer of knowledge develops deeper student learning as compared to memorizing an algorithm (Math Practice 7). 

I may walk either the class as a whole, or individual students, through finding the decision of the type of compound inequality and any scaffolding (pace or hints) will depending on the class and/or individual student(s) needs.  Some students may discover the algorithm independently, in which case I would encourage them to share with the class.  Once students have solved the inequality, I encourage them to check their work graphically.  As before, details are found in the PowerPoint- Absolute Value Inequalities.

I have also included some real-life scenarios using gas mileage and test scores.  The first is similar to one found in my absolute value equations lesson.  The second problem gives the maximum and minimum of an absolute value rather than the center and the range and asks the students to create the absolute value inequality (Math Practice 4).  I would recommend the Note Card Activity, if there is time, particularly for the second scenario.

Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

This Exit Ticket asks students to solve a multi-step absolute value inequality. 


The goal of this Homework is for students to reinforce the skills learned in the day’s lesson.  The first six problems ask students to solve absolute value inequalities of varying difficulty.  The next two problems provide modeling opportunities.  The final question ask student to describe what the graph of an absolute value equation with one real solution and one extraneous solution would look like (Math Practice 7).