Unit 6

Efficiently Solving Inequalities

Expressions, Equations, and Inequalities

Lesson 15

Expressions and Equations

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Let’s solve more complicated inequalities.

Unit 6 ● Lesson 15

Learning

Goal

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Lots of Negatives

Unit 6 ● Lesson 15 ● Activity 1

Here is an inequality: -x ≥ -4.

1. Predict what you think the solutions on the number line will look like.
2. Select all the values that are solutions to -x ≥ -4:
1. 3
2. -3
3. 4
4. -4
5. 4.001
6. -4.001
3. Graph the solutions to the inequality on the number line:

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Inequalities with Tables

Unit 6 ● Lesson 15 ● Activity 2

• How are the numbers in the top row and bottom row related?
• Think about the equation x + 2 = -2. What value of x makes this true? Where do you see that in the table?
• Think about the inequality x + 2 > 3. What values of x make this true? Where do you see that in the table?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Inequalities with Tables

Unit 6 ● Lesson 15 ● Activity 2

1. Let's investigate the inequality x – 3 > -2.

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• Complete the table.
• For which values of x is it true that x – 3 = -2 ?
• For which values of x is it true that x – 3 > -2?
• Graph the solutions to x – 3 > -2 on the number line:

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Inequalities with Tables

Unit 6 ● Lesson 15 ● Activity 2

• Here is an inequality: 2x < 6.
• Predict which values of x will make the inequality 2x < 6 true.
• Complete the table. Does it match your prediction?

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• Graph the solutions to 2x < 6 on the number line:

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Inequalities with Tables

Unit 6 ● Lesson 15 ● Activity 2

• Here is an inequality: -2x < 6.
• Predict which values of x will make the inequality -2x < 6 true.
• Complete the table. Does it match your prediction?

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• Graph the solutions to -2x < 6 on the number line:

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• How are the solutions to 2x < 6 different from the solutions to �-2x < 6?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Inequalities with Tables

Unit 6 ● Lesson 15 ● Activity 2

• How does solving the associated equation to an inequality help you solve the inequality?
• Do you need to complete the entire table to find out on which side of the boundary value the solutions are?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Which Side are the Solutions?

Unit 6 ● Lesson 15 ● Activity 3

1. Let’s investigate -4x + 5 ≥ 25.
1. Solve -4x + 5 = 25.
2. Is -4x + 5 ≥ 25 true when x is 0? What about when x is 7? What about when x is -7?
3. Graph the solutions to -4x + 5 ≥ 25 on the number line.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Which Side are the Solutions?

Unit 6 ● Lesson 15 ● Activity 3

• Let’s investigate .
• Solve .
• Is true when x is 0?
• Graph the solutions to on the number line.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Which Side are the Solutions?

Unit 6 ● Lesson 15 ● Activity 3

• Solve the inequality 3(x + 4) > 17.4 and graph the solutions on the number line.

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4. Solve the inequality and graph the solutions on the number line.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Efficiently Solving Inequalities

Unit 6 ● Lesson 15

• What if someone asked for your help with how to solve inequalities?
• What would you tell them?
• How would you describe to someone how to solve any inequality?
• Write down your ideas. Be ready to share your thoughts.

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Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Unit 6 ● Lesson 15

• I can graph the solutions to an inequality on a number line.
• I can solve inequalities by solving a related equation and then checking which values are solutions to the original inequality.

Learning

Targets

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Testing for Solutions

Unit 6 ● Lesson 15 ● Activity 4

For each inequality, decide whether the solution is represented by x < 2.5 or x > 2.5.

1. -4x + 5 > -5

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2. -25 > -5(x + 2.5)

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Cool-down

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

solution to an inequality

Unit 6 ● Lesson 15

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

For example, 5 is a solution to the inequality c < 10, because it is true that 5 < 10. Some other solutions to this inequality are 9.9, 0, and -4.

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Glossary

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/.

All curriculum excerpts are under the following licenses:

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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