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Unit 8, Lesson 16

Solving Inequalities with Absolute Values

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Integrated Math 1

16.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

In this Warm-up, students use their understanding of the connection between absolute value equations and distances on a number line to describe the connection for inequalities. This will be useful in later activities when students use the connection to reason about solutions to inequalities involving absolute values.

Pay attention to the precision of student language as they describe the inequalities (MP6). In particular, phrases like “at most” and “at least” usually imply that the endpoints are included, while phrases like “less than” or “greater than” usually imply the endpoints are not included.

Launch

Arrange students in groups of 2.

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “Describe what each of these expressions mean.” Invite listeners to ask questions, and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening

Activity

None

Describe what each of these expressions mean.

Activity Synthesis

The purpose of this discussion is to ensure students understand inequalities involving absolute value as a range of distances. For each question, select a group to share their solution.

Ask students,

"What is the difference between these two sentences? 'The distance between and 8 is less than 4' and 'The distance between and 8 is at most 4.'" (The first sentence would use the symbol and the second sentence would use the symbol. In the first sentence, 4 is not included as a distance that works. In the second sentence it is included.)
"How would you write an inequality that means, 'all the values that are less than 5 away from 10?'" ()

16.2

Activity

Instructional Routines

MLR6: Three Reads

Materials

To Gather

Nut and bolt

Activity Narrative

In this activity, students move from writing individual values that satisfy conditions for a real-world situation to writing inequalities that describe the conditions. The context of nuts and bolts is used to emphasize the importance of precision and give students a sense of how error tolerance is used in the real world.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Launch

Arrange students in groups of 2.

While an image is provided, it can be helpful to show students a bolt with a nut that fits it so that students can examine how the threads on each work together.

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the image in the problem stem, without revealing the questions.

For the first read, read the problem aloud, then ask, “What is this situation about?” (the thread distance on a bolt). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (the ideal distance of threads on a bolt, the amount of error that is acceptable)
After the third read, reveal the question: "What are 3 different distances between threads that would be acceptable for this type of bolt?" and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (think of what numbers might be within 0.1 mm of 1 mm)

Representation: Access for Perception. Use a bolt and nut to demonstrate the situation, and allow students to touch the threads.
Supports accessibility for: Conceptual Processing, Language, Memory

16.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students practice reasoning about distances on the number line to solve inequalities that include absolute value.

Launch

Keep students in groups of 2.

Remind students of how to graph solutions to inequalities on a number line. Pay particular attention to the closed or open circle based on the inclusivity of the inequalities. Here are some examples.

A number line shaded to the left of an open circle at 4.

A number line shaded to the right of a closed dot at -3.

Lesson Synthesis

The purpose of the discussion is to ensure students make the connection between absolute value inequalities and distances on a number line.

Ask students,

“One way to refer to all of the numbers from 0 to 10 is to talk about numbers that are within 5 of the number 5. How could you write this as an inequality that uses absolute values?” ( or , depending on if we want to include 0 and 10 or not.)
“To be safe, you should stand more than 6 feet away from a small fire when using a fire extinguisher. If the fire is at a position that is considered the 10 foot mark, how can you write this as an inequality using absolute value?” ()
“To be effective, you should stand no more than 10 feet away from a small fire when using a fire extinguisher. If the fire is still at a position that is considered the 10 foot mark, how can you write this as an inequality using absolute value?” ()
“When an inequality looks like for numbers and , is the solution a single shaded region or two outside regions? Explain your reasoning.” (a single shaded region, because the distance from to a number is less than , so all of the numbers that work should be within a certain range of )

to access Cool-down.

Student Lesson Summary

In many real-world situations, it is okay if an item is not exactly the perfect length or weight. There is a range of values that will work: The length or weight of the item can be within a certain amount or at least a certain amount away. In these cases, it can be useful to write the range of values that work as inequalities that include absolute values.

To be within a certain distance of a value, we can write an inequality of the form . The solutions to this inequality can be drawn on a number line like this:

A number line labeled a minus b, a, and a plus b. The region from a minus b to a plus b is shaded with the endpoints indicated by open circles.

To be at least a certain distance away from a value, we can write an inequality of the form . The solutions to this inequality can be drawn on a number line like this:

A number line marked with a minus b, a, and a plus b. It is shaded from the left of the number line to a closed dot a a minus b and from a closed dot at a plus b and to the right.

For example, to fence in a circular area with a radius of 100 meters, we should use the equation and buy meters of fencing. This is difficult to do because is an irrational number. Maybe it’s okay if we don’t make a perfect circle or if it’s a little off, so if we buy amount of fence, where is a solution to , we will have the right amount of fencing within 1 meter of the exact value, and that should be close enough.

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Lesson 16

Solving Inequalities with Absolute Values

Warm-up

Standards Alignment

Building On

Addressing

HSA-CED.A.1

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-CED.A.1

HSA-REI.B.3

Building Toward

Standards Alignment

Building On

Addressing

HSA-REI.B.3

Building Toward