Curriculum  /  Math  /  11th Grade  /  Unit 4: Rational and Radical Functions  /  Lesson 16

Rational and Radical Functions

Lesson 16

Math

Unit 4

11th Grade

Lesson 16 of 18

Objective

Solve rational equations.

Common Core Standards

Core Standards

  • A.APR.D.6 — Rewrite simple rational expressions in different forms; write a(x /b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
  • A.REI.A.2 — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  • 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

  • A.REI.A.1

Criteria for Success

  1. Apply the rules of common denominators to solve rational equations. 
  2. Describe how when the denominators are equal, you can just solve the equation with the numerators. 
  3. Identify extraneous solutions and explain how these are not solutions to the rational equation.
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Anchor Problems

25-30 minutes

Problem 1

Solve the following equation: 

 

$${{3\over{x}}={8\over{x-2}}}$$

 

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 26 — Exercise 3

Algebra II > Module 1 > Topic C > Lesson 26 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Megan is solving this equation:

 

$${{2\over{x^2-1}}-{1\over{{x-}1}}={1\over{x+1}}}$$

 

She says:

If I clear the denominators, I find that the only solution is $${{x=1}}$$, but when I substitute in $${{x=1}}$$ the equation doesn’t make any sense. 

  1. Is Megan’s work correct? 
  2. Why does Megan’s method produce an $${x-}$$value that does not solve the equation? 

Guiding Questions

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References

Illustrative Mathematics An Extraneous Solution

An Extraneous Solution, accessed on Nov. 19, 2017, 6:19 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

5-10 minutes

Find all solutions to the following equation. If there are any extraneous solutions, identify them and explain why they are extraneous. 

 

$${{7\over{b+3}}+{5\over{b-3}}={10b\over{b^2-9}}}$$

 

References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 26 — Exit Ticket

Algebra II > Module 1 > Topic C > Lesson 26 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

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 problems where every term is not already a fraction. For example: $${x-{6\over{x}}=5}$$  or  $${{15\over{4}}={6\over{x+2}}+3}$$.

Next

Write and solve rational functions for contextual situations.

Lesson 17
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Lesson Map

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

Topic A: Introduction to Rational and Radical Functions and Expressions

Topic B: Features of Rational Functions and Graphing Rational Functions

Topic C: Solve Rational and Radical Equations and Model with Rational Functions

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