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

Functions and Transformations

Lesson 9

Math

Unit 5

9th Grade

Lesson 9 of 16

Objective

Identify the solution(s) to an absolute value inequality.

Common Core Standards

Core Standards

  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 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.
  • F.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  • 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.B.3 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • 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

  1. Understand an absolute value inequality $${ {|x|<a}}$$ or $${{|x|>a}}$$ as a system of two functions compared to one another (i.e., $${f(x)<g(x)}$$) where one function is an absolute value function and one function is a constant function.
  2. Understand the solution to $${|x|<a}$$ or $${{|x|>a}}$$ is the values for $$x$$ where the function $$f(x)=|x|$$ is less than or greater than the function $$g(x)=a$$ in the coordinate plane. 
  3. Solve simple absolute value inequalities algebraically and verify the solutions graphically.

Tips for Teachers

Similar to Lesson 7, this lesson introduces the concept of absolute value inequalities by looking at functions in the coordinate plane.

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Anchor Problems

25-30 minutes

Problem 1

Consider the three situations shown below.

$${|x|=5}$$
$${|x|<5}$$
$${|x|>5}$$

For each situation, let $${{f(x)}=|x|}$$ and $${{g(x)}=5}$$. Graph each relationship between $${f(x)}$$ and $${g(x)}$$ in the coordinate plane and then use each graph to determine the solution to each situation. 

Guiding Questions

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Problem 2

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

a.   $${\left|x-{1\over2}\right|<3}$$

b.   $${\left|2x\right|>{7\over2}}$$

c.   $${\left|{1\over3}x\right|-4<-1}$$

 

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

Solve each absolute value inequality algebraically and match it to a solution or graph shown on the right. 

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.

  • Algebra By Example 5.4 Compound Inequalities — (Review of solving compound inequalities; does not include absolute value)
  • Kuta Software Free Algebra 1 Worksheets Absolute Value Inequalities — #1-6 and #11-16 (Ask students to graph a few examples to verify algebraically)

Next

Solve absolute value inequalities algebraically and verify graphically.

Lesson 10
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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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