Lesson 20B: Absolute Value Equations and Inequalities

Hart Interactive – Algebra 1 M1 Lesson 20B ALGEBRA I

Lesson 20B: Absolute Value Equations and Inequalities

Warm-Up Exercise

1. Watch the absolute value video on YouTube Math Shorts Episode 10 and then answer the questions

below. https://www.youtube.com/watch?v=wrof6Dw63Es

A. 1.3 ______− = B. 4.75 ______= C. 10 4 ______− + − = D. 11 21 ______− − − =

E. if 4.5, then _____or_____x x= = F. if 3 6, then _____or_____x x+ = =

2. Use the number lines below to graph each inequality.

A. x < 4 B. x > -1 C. 2 + x < 5


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Exploratory Exercise

First, let’s look at absolute value equations, like we saw in Exercise 1E and 1F.

3. For each absolute value equation below, think about any values of x that will make the equation true. Are

there two solutions for each one?

A. 2 4x − = B. 2 4x − = C. 2 4x = D. 4x = −

4. Kyle wrote the following steps to solve absolute value equations.

1. Get the absolute value term alone. You can add, subtract, multiply or divide to get it alone.

2. You have to make two equations. Take what is in the absolute value symbol and make it equal to the number on the other side of the equal sign and also take that same inside part and set it equal to the opposite of the number on the other side of the equal sign.

3. Solve each equation.

4. Check your work.

A. Use Kyle’s steps to solve 1 2 7x + − = .

B. What other steps should be added to Kyle’s steps to cover all absolute value situations?


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5. Play the Who Wants to be a Hundredaire Game and

record your work and answers below. The questions

below are mixed up so you’ll have to look carefully for

the one that is being played on the video. There are two

extra problem below that are not on the video.

http://www.math-play.com/Absolute-Value-Equations/Absolute-Value-Millionaire.html

A. 9 2k− + = − B. 2 12x = C. 2 3 24x− =

D. 2 10 14x + = E. 2 4 14x − = F. 1 7n+ =

G. 2 1x− + = H. 10 10 50n− = I. 6 7a− =

J. 3 48x− = K. 4 3n÷ = L. 5 10 15y − =


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Next, we’ll combine two ideas – inequalities and absolute value.

6. For each inequality below, think about all the points that would make the inequality true and plot those

points. What type of endpoint should each inequality have? Are there other points that would also make

the inequality true?

A. 4x < B. 1x ≥ − C. 1 3x+ ≤

Tanja says that there is a better way to get the solutions, rather than just guessing. Below are her steps.

Think “greatOR”!

Isolate the absolute value expression on

the left side of the inequality.

Your equation either has no solution or all real numbers as solutions. Use the sign of each

side of your inequality to decide

which of these cases holds.

Remove the absolute value bars by setting up a

compound inequality. The type of inequality sign in

the problem will tell us how to set up the compound

inequality.

(stuff inside absolute value) > (number)

OR

(stuff inside absolute value) < - (number)

- (number) < (stuff inside absolute value) < (number)

Think “less thAND”!


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5. A. Follow Tanja’s steps for the absolute value inequality 2 3 1 20x − < . Be sure to show each step.

B. Graph the solution set on the number line.

6. A. Follow Tanja’s steps for the absolute value inequality 1

2 3 6 52

x + + < . Be sure to show each step.


7. A. Follow Tanja’s steps for the absolute value inequality 3 2 6 5x + + ≥ . Be sure to show each step.


C. Does Tanya’s steps work for < or > absolute value problems? Explain.


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Compare and Contrast Kyle and Tanja’s Absolute Value Steps 8. Work with your partner to find at least two true statements for each section of the Venn diagram. In the

overlapping section, write two statements that are true for both procedures.

Kyle’s Absolute Value Steps

Tanja’s Absolute Value Steps


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Homework Problem Set Solve each absolute value equation. Write your answer in set notation.

1. 6 10k − = 2. 6 13a + =

3. 104

8

n− +=

4. 3 27r− =

5. 2

4

n=

6. 6 7 4 38r− + =

7. 8 10 6 22p− + − = − 8. 10 3 2 22r+ − =

9. 2 8 7 2 42k+ − = 10. 7 3 5 7 0n+ − =


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Solve and graph each absolute value inequality.

11. 1r <

12. 10 15p− <

13. 9 5n − ≥ −

14. 1 0

10

r− + >

15. 5 13k − >

16. 4 72

3

n−>

17. 10 7 97p− − > −

18. 8 5 6 7 49m− + − ≤


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19. Lindsey is making some home-made toffee. The recipe says that she must bring the mixture to a boil at

285 degrees. If she is 7 degrees above or below, the toffee should turn out fine.

A. Write and solve an absolute value equation to model the minimum and maximum temperatures that

would still create yummy toffee.

B. Write, solve, and graph an absolute value inequality to model the range of temperatures that will

make yummy toffee.

CHALLENGE PROBLEMS 20 – 22

20. Solve for x using the inequality a x b c d− + ≤ . Assume that d – c > 0.


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21. Write a simple inequality with an absolute value symbol whose solution would be represented by the

graph shown below.

22. A student made an error in the following problem. Determine where the error was made and then

complete the problem correctly.

3 9 5

3 4

4 3 4

1 7

x

x

x

x

+ + <

+ < −

< + < −

< < −


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Documents

Lesson 20B: Absolute Value Equations and Inequalities