Unit 8, Lesson 15
Solving Equations with Absolute Values
Integrated Math 1
15.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students look at how adding or subtracting a value in different places in an absolute value function changes the graph of the function. First, they match a function to its graph, then to a description based on a familiar context. Monitor for students who:
- Check coordinate pairs or input values.
- Reason about transforming the graph or finding distance.
Launch
Arrange students in groups of 2.
Tell students to close their books or devices (or to keep them closed). Display the graph for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph.
If the connection to absolute value does not come up during the conversation, ask students to discuss this idea.
Activity
None- Which of these functions is represented by this graph? Explain your reasoning.
- One of these functions takes a guess as an input, , and outputs the distance between the guess and 30 as the output. Which function is it? Explain your reasoning.
- Write a function that takes a guess as an input and outputs the distance between the guess and -8.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of the discussion is to connect the different representations to the function involving absolute value.
Invite students to share their solutions to the first two questions, including their reasoning. If possible, include at least 1 group that checked values to find the answer and 1 group that reasoned about the structure of the function.
Then invite a group to share their answer for the last question. Ask students what the graph of this function would look like (a V shape with the point at ).
Student Lesson Summary
The statement is true and can be interpreted as "The distance between 3 and 5 is 2." This is why it is also true to write .
The idea can also be used to solve equations in which a distance is known to a point. For example, tells us that the distance between and 4 is 6. On a number line, it might look like this.
What values of make this statement true? Both and make the equation true, so they are solutions to the absolute value equation.
Another way to solve an equation involving an absolute value is to look at the piecewise meaning.
We can take the equation and split it into two cases.
- If , then the equation becomes . We can solve this using what we learned about linear equations. Adding 4 to each side, we can rewrite the equation as . We can check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is greater than or equal to 4.
- If , then the equation becomes . We can distribute the -1 on the left to get . From there, we can subtract 4 from each side and multiply each side of the result by -1 to get . We can also check that this is fine for this case because the case begins with "If . . ." and ends with a value for that is less than 4.
Putting the results of the cases together, we again get that the solutions are -2 and 10.
Whether you prefer reasoning about distances on a number line or using the piecewise definition to write equations in different cases, the solutions should be the same.