Conquering Rational Equations: Clearing Denominators & Avoiding Extraneous Solutions

Algebra 2 Grades High School 2:11 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to solve rational equations by clearing denominators and identifying extraneous solutions, ensuring accurate results. This lesson builds upon basic algebra skills and extends them to more complex rational expressions.

Video Resource

Solving Rational Equations (Clearing the Denominator)

Mario's Math Tutoring

Duration: 2:11
Watch on YouTube

Key Concepts

  • Rational Equations: Equations containing rational expressions (fractions with polynomials).
  • Clearing Denominators: Multiplying all terms in an equation by the least common denominator (LCD) to eliminate fractions.
  • Extraneous Solutions: Solutions obtained algebraically that do not satisfy the original equation (often due to making a denominator zero).

Learning Objectives

  • Students will be able to identify rational equations.
  • Students will be able to solve rational equations by clearing denominators.
  • Students will be able to identify and discard extraneous solutions.

Educator Instructions

  • Introduction (5 mins)
    Begin by defining rational equations and reviewing the concept of common denominators. Briefly discuss why clearing denominators is a useful strategy for solving these equations. Introduce the idea of extraneous solutions and their potential impact on the solution set.
  • Video Explanation (10 mins)
    Play the Mario's Math Tutoring video 'Solving Rational Equations (Clearing the Denominator)'. Instruct students to pay attention to the steps involved in clearing denominators, solving the resulting equation, and checking for extraneous solutions. Pause at key points (e.g., after solving for x in each example) to allow for student questions and clarification.
  • Worked Example 1: Clearing Denominators (10 mins)
    Work through a simple example similar to the one in the video (3/x - 2/4 = 8/(4x)). Emphasize the process of finding the common denominator (4x), multiplying each term by it, simplifying, and solving for x. Stress the importance of checking the solution by substituting it back into the original equation. Identify any potential extraneous solutions.
  • Worked Example 2: Identifying Extraneous Solutions (10 mins)
    Present a more complex equation, such as 21/(x-4) - 16/(x-3) = (5x+1)/(x^2 - 7x + 12). Guide students through factoring the quadratic denominator, finding the LCD, clearing denominators, and solving the resulting equation. Emphasize the need to check ALL solutions in the original equation to identify any extraneous solutions. Explain why values that make any denominator zero are extraneous.
  • Independent Practice (10 mins)
    Provide students with 2-3 rational equations to solve independently. Encourage them to follow the steps outlined in the video and worked examples. Circulate to provide assistance as needed. After the practice, select students to showcase their work on the board.

Interactive Exercises

  • Error Analysis
    Present students with worked examples containing common errors in solving rational equations (e.g., incorrect LCD, forgetting to distribute, failing to check for extraneous solutions). Have them identify and correct the errors.
  • Create Your Own
    Challenge students to create their own rational equations, solve them, and identify any extraneous solutions. They can then exchange equations with a partner to solve.

Discussion Questions

  • What is a rational equation and how does it differ from a linear or quadratic equation?
  • Why is it important to check for extraneous solutions when solving rational equations?
  • What strategies can be used to find the least common denominator (LCD) of multiple rational expressions?
  • Explain in your own words the steps required to solve a rational equation.

Skills Developed

  • Solving equations
  • Simplifying rational expressions
  • Identifying and avoiding extraneous solutions
  • Critical thinking

Multiple Choice Questions

Question 1:

What is the first step in solving a rational equation?

Correct Answer: Find the least common denominator (LCD)

Question 2:

What is an extraneous solution?

Correct Answer: A solution that makes a denominator equal to zero

Question 3:

When checking for extraneous solutions, where should you substitute your answers?

Correct Answer: The original equation

Question 4:

What is the least common denominator (LCD) of 1/x and 1/(x+2)?

Correct Answer: x(x+2)

Question 5:

If you solve a rational equation and find x = 3, but x = 3 makes a denominator zero in the original equation, what does that mean?

Correct Answer: x = 3 is an extraneous solution

Question 6:

Solve for x: 4/x = 2/5

Correct Answer: x = 10

Question 7:

Which of the following is an example of a rational equation?

Correct Answer: 3/x + 2 = 1

Question 8:

Which is a step to clear a rational equation?

Correct Answer: Multiply each term with the LCD

Question 9:

What is the solution to the equation x/3 = 5/x?

Correct Answer: x = ±√15

Question 10:

Solve for x: 1/x = 1/(x-1)

Correct Answer: x = Does Not Exist

Fill in the Blank Questions

Question 1:

A(n) _________ solution is a solution obtained algebraically that does not satisfy the original equation.

Correct Answer: extraneous

Question 2:

To solve a rational equation, you first need to clear the _________.

Correct Answer: denominators

Question 3:

The abbreviation LCD stands for _________ _________ _________.

Correct Answer: least common denominator

Question 4:

If a solution makes a denominator equal to zero, it is considered _________.

Correct Answer: extraneous

Question 5:

Before solving a rational equation, you may need to _________ the denominator to find the LCD.

Correct Answer: factor

Question 6:

A rational equation involves two _________

Correct Answer: fractions

Question 7:

To ensure you found all correct answers, you must _________

Correct Answer: check

Question 8:

Solve for x: x/4 + 1 = 5; x = _________

Correct Answer: 16

Question 9:

Factor x^2 + 6x + 9; (x + _________)^2

Correct Answer: 3

Question 10:

Solve for x: 1/2x = 5/4; x = _________

Correct Answer: 2/5

Educational Standards

A2.A.1.3:Solve one-variable rational equations and check for extraneous solutions. A2.A.2.3:Add, subtract, multiply, divide, and simplify rational expressions. A2.A.2.4:Solve rational equations with real roots.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans