Curriculum  /  Math  /  9th Grade  /  Unit 5: Functions and Transformations  /  Lesson 7

Functions and Transformations

Lesson 7

Math

Unit 5

9th Grade

Lesson 7 of 16

Objective

Identify the solutions to an absolute value equation.

Common Core Standards

Core Standards

  • A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.EE.C.7
  • 8.EE.C.8

Criteria for Success

  • Understand an absolute value equation $${{{|x|=a}}}$$ as a system of two functions set equal to one another (i.e., $${f(x)=g(x)}$$) where one function is an absolute value function and one function is a constant function.
  • Understand the solution to $${{{|x|=a}}}$$ is the value(s) for $$x$$ where the function $${f(x)=|x|}$$ intersects function $${g(x)=a}$$ in the coordinate plane. 
  • Identify situations where $${{{|x|=a}}}$$ has two solutions, one solution, or no solutions.
  • Solve simple absolute value equations algebraically and verify the solutions graphically.

Tips for Teachers

Students see a graphical representation of $${ |x|=a}$$ in the coordinate plane as opposed to a number line. See the notes for Anchor Problem #2 for a visual example of an absolute value equation with two, one, and no solutions in the coordinate plane.

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Anchor Problems

25-30 minutes

Problem 1

Consider the equation $${ {|x|=5}}$$. 

This can be thought of as a system of equations, where

$${f(x)=\left | x \right |}$$
$${{g(x)}=5}$$
and $${f(x)={g(x)}}$$

 

Graph $${f(x) }$$ and $${g(x)}$$ in the coordinate plane. Based on the graph, what are the solutions to $${|x|=5}$$?

Guiding Questions

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Problem 2

For each absolute value equation below, answer the questions that follow. 

$${|x|=-4}$$                     $${ |x|=0}$$                    $${ |x|=4}$$

 

  1. How many solutions are there for $$x$$ in each equation?
  2. Thinking of each equation as two equal functions (similar to Anchor Problem #1), what does each one look like in the coordinate plane? Where do you see the solution(s)?

Guiding Questions

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Problem 3

Solve each absolute value equation algebraically and verify the solution graphically. 

$${|x+2|=8}$$

$${|x|+2=8}$$

$${|x+2|+1=0}$$

Guiding Questions

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Problem Set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

5-10 minutes

Problem 1

Determine if each equation has two, one, or no solutions. Put a check in the appropriate column in the table. 

  2 solutions 1 solution No solutions
$${|x|+3=3}$$      
$${|x-3|=-3}$$      
$${|3x|=3}$$      

Problem 2

Find the solution(s) to the absolute value equation algebraically. Then sketch a graph in the coordinate plane to verify the solution(s).

$${|x-4|=3}$$

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include examples where students write their own absolute value equations with 2, 1, or no solutions.
  • Kuta Software Free Algebra 1 Worksheets Absolute Value Equations — #1-12 (Have students verify the solutions graphically using technology)

Next

Solve absolute value equations algebraically and identify extraneous solutions.

Lesson 8
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Piecewise Functions

Topic B: Absolute Value Functions

Topic C: Function Transformations

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