Lesson Plan: Adding and Subtracting Rational Expressions

Lesson Summary

This lesson introduces students to adding and subtracting rational expressions. Students will learn how to find common denominators, rewrite expressions with equivalent fractions, and simplify the final result. The lesson includes step-by-step examples and real-world applications to reinforce conceptual understanding. Multimedia resources from Media4Math.com provide visual support and interactive engagement throughout the lesson. This lesson is designed for a 50-minute class period and concludes with a 10-question quiz with an answer key.

Lesson Objectives

• Identify and find the least common denominator (LCD) for rational expressions.
• Rewrite rational expressions using the LCD.
• Add and subtract rational expressions.
• Simplify results and check for domain restrictions.

Common Core Standards

• CCSS.MATH.CONTENT.HSA.APR.D.7 – Understand addition and subtraction of rational expressions.

Prerequisite Skills

• Finding least common multiples.
• Adding and subtracting fractions.

Key Vocabulary

• Common Denominator - A shared multiple of the denominators of two or more fractions.
  • Multimedia Resource: https://www.media4math.com/library/74761/asset-preview
• LCD (Least Common Denominator) - The smallest common multiple of the denominators in a set of fractions.
  • Multimedia Resource: https://www.media4math.com/library/43138/asset-preview
• Simplify - To rewrite an expression in its simplest form by reducing fractions and combining like terms.

Multimedia Resources from Media4Math.com

• Video: "Adding Fractions Using Equivalent Fractions" - https://www.media4math.com/library/39462/asset-preview
• Slide Show: Animated Clip Art of Rational Expressions: https://www.media4math.com/library/slideshow/slide-show-animated-clip-art-rational-expressions
• A collection of animated clip art on Rational Expressions: https://www.media4math.com/MathClipArt--AnimatedClipArtRationalExpressions

Warm-Up Activities

Choose from one or more of these activities:

Activity 1: Review Least Common Multiple (LCM)

Students will review how to find the Least Common Multiple (LCM) of two numbers, a skill necessary for finding the Least Common Denominator (LCD) in rational expressions.

1. Write the following pairs of numbers on the board: 4 and 6, 5 and 10, 7 and 14.
2. Ask students to find the LCM of each pair.
3. Have volunteers explain their solutions step by step.
4. Introduce a more challenging pair: 12 and 18 to reinforce the concept.

Discuss why finding the LCM is useful when working with fractions.

Activity 2: Adding and Subtracting Numerical Fractions

Before working with rational expressions, students will practice adding and subtracting numerical fractions.

1. Write the following fractions on the board:
   • \( \frac{2}{5} + \frac{3}{10} \)
   • \( \frac{5}{6} - \frac{1}{4} \)
2. Ask students to identify the LCD for each pair of fractions.
3. Guide them through rewriting the fractions with the LCD and solving.
4. Have students share their work with a partner to compare answers.

Emphasize that the same process will apply to rational expressions.

Activity 3: Group Activity - Identifying LCDs for Algebraic Fractions

Students will work in small groups to identify LCDs for rational expressions.

1. Divide students into small groups.
2. Assign each group a different rational expression to work with:
   • \( \frac{1}{x} + \frac{1}{2x} \)
   • \( \frac{3}{x} - \frac{2}{3x} \)
   • \( \frac{5}{4x} + \frac{7}{3x} \)
3. Have each group determine the LCD and rewrite the fractions accordingly.
4. Each group presents their solution to the class.

Discuss common mistakes and strategies for finding the LCD efficiently.

Teach

Introduction (5 minutes)

Before adding and subtracting rational expressions, recall that rational expressions follow the same rules as numerical fractions.

• Fractions must have a common denominator before they can be added or subtracted.
• The Least Common Denominator (LCD) is the smallest multiple that both denominators share.
• Once fractions have the same denominator, their numerators can be combined.

Examples of fraction addition:

 Now we will extend this process to rational expressions.

Example 1: Adding Rational Expressions with Like Denominators

Given the rational expressions:

\[ \frac{3}{x+1} + \frac{5}{x+1} \]

Since the denominators are already the same:

\[ \frac{3+5}{x+1} = \frac{8}{x+1} \]

Key Takeaway: When denominators are identical, simply add the numerators.

Domain Restriction: \( x \neq -1 \) (since the denominator cannot be zero).

Example 2: Adding Rational Expressions with Unlike Denominators

Consider:

\[ \frac{2}{x} + \frac{3}{x+2} \]

Step 1: Find the LCD. The denominators are \( x \) and \( x+2 \), so the LCD is:

\[ x(x+2) \]

Step 2: Rewrite each fraction using the LCD:

\[ \frac{2(x+2)}{x(x+2)} + \frac{3x}{x(x+2)} \]

Step 3: Expand the numerators:

\[ \frac{2x + 4}{x(x+2)} + \frac{3x}{x(x+2)} \]

Step 4: Combine the numerators:

\[ \frac{2x + 4 + 3x}{x(x+2)} = \frac{5x+4}{x(x+2)} \]

Domain Restrictions: \( x \neq 0, -2 \) (denominators cannot be zero).

Example 3: Subtracting Rational Expressions

Given:

\[ \frac{x+3}{x^2-4} - \frac{2}{x+2} \]

Step 1: Factor the denominator of the first fraction:

\[ x^2 - 4 = (x-2)(x+2) \]

Step 2: Identify the Least Common Denominator (LCD):

\[ (x-2)(x+2) \]

Step 3: Rewrite each fraction with the LCD:

\[ \frac{x+3}{(x-2)(x+2)} - \frac{2(x-2)}{(x-2)(x+2)} \]

Step 4: Expand the numerator:

\[ \frac{x+3 - (2x-4)}{(x-2)(x+2)} \]

Step 5: Simplify the numerator:

\[ \frac{x+3 - 2x + 4}{(x-2)(x+2)} \] \[ \frac{-x+7}{(x-2)(x+2)} \]

Final Answer: \( \frac{7-x}{(x-2)(x+2)} \) (Rewriting with positive coefficient)

Domain Restrictions: \( x \neq \pm 2 \).

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Example 4: Real-World Application - Water Tank Problem

A large water tank is being filled by two pipes, each supplying water at a different rate. The first pipe fills the tank at a rate of \( \frac{3}{x+2} \) gallons per minute, and the second pipe fills it at a rate of \( \frac{2}{x} \) gallons per minute.

Context:

• \( x \) represents the number of minutes after the pipes are opened.
• The rational expressions measure the rate at which water is flowing into the tank.
• Objective: Find the total water flow rate when both pipes are open simultaneously.

Step 1: Find the LCD.

The denominators are \( x+2 \) and \( x \), so the Least Common Denominator (LCD) is:

\[ x(x+2) \]

Step 2: Rewrite each fraction using the LCD:

\[ \frac{3x}{x(x+2)} + \frac{2(x+2)}{x(x+2)} \]

Step 3: Expand the numerators:

\[ \frac{3x + 2x + 4}{x(x+2)} \] \[ \frac{5x + 4}{x(x+2)} \]

Final Answer: \( \frac{5x+4}{x(x+2)} \).

Interpretation:

• The total rate at which water enters the tank is represented by \( \frac{5x+4}{x(x+2)} \) gallons per minute.
• This expression shows how the combined flow rate changes over time as \( x \) increases.
• If \( x \) is very large, the total flow rate approaches \( \frac{5}{x} \), meaning the water flows more slowly over time.

Domain Restrictions: \( x \neq 0, -2 \), since the pipes' rates depend on time values that must be positive.

Example 5: Real-World Application - Sneaker Sales Revenue

A sportswear company has launched two lines of sneakers: Urban Runners and Elite Trainers. The company tracks revenue from each line over time.

Based on past sales trends, the revenue per month for each sneaker line is modeled by:

\[ \frac{5}{x+3} + \frac{2}{x} \]

Assumptions:

• \( x \) represents the number of months since the sneakers were launched.
• The company sees a gradual increase in sales as awareness grows.
• \( \frac{5}{x+3} \) represents the revenue (in thousands of dollars per month) from Urban Runners, which started three months before the official launch.
• \( \frac{2}{x} \) represents revenue from Elite Trainers, which launched exactly at month 0.
• Objective: Find the total revenue function combining both sneaker lines.

Step 1: Identify the LCD.

The denominators are \( x+3 \) and \( x \), so the Least Common Denominator (LCD) is:

\[ x(x+3) \]

Step 2: Rewrite each fraction using the LCD:

\[ \frac{5x}{x(x+3)} + \frac{2(x+3)}{x(x+3)} \]

Step 3: Expand the numerators:

\[ \frac{5x + 2x + 6}{x(x+3)} \] \[ \frac{7x + 6}{x(x+3)} \]

Final Answer: \( \frac{7x+6}{x(x+3)} \).

Interpretation:

• The total revenue function is represented by \( \frac{7x+6}{x(x+3)} \), showing how revenue from both sneaker lines accumulates over time.
• For small values of \( x \), revenue fluctuates more due to initial marketing effects and new customer adoption.
• For large values of \( x \), the function approaches \( \frac{7}{x} \), meaning the monthly revenue stabilizes as demand levels off.
• The domain restriction \( x \neq 0, -3 \) means the function does not apply at the launch moment for Elite Trainers (\( x=0 \)) or before Urban Runners existed (\( x=-3 \)).

Business Insight:

• The company can use this function to forecast total revenue and adjust inventory and marketing efforts accordingly.
• If Urban Runners sales slow down while Elite Trainers grow, the function would need to be adjusted to reflect changing consumer demand.

Review

Key Vocabulary Review

Before practicing additional examples, let's review the key vocabulary from this lesson.

• Common Denominator - A shared multiple of the denominators of two or more fractions.
• LCD (Least Common Denominator) - The smallest common multiple of the denominators in a set of fractions.
• Simplify - To rewrite an expression in its simplest form by reducing fractions and combining like terms.

Example 1: Subtracting Rational Expressions

Given:

\[ \frac{4}{x+2} - \frac{2}{x} \]

Step 1: Find the LCD. The denominators are \( x+2 \) and \( x \), so the LCD is:

\[ x(x+2) \]

Step 2: Rewrite each fraction using the LCD:

\[ \frac{4x}{x(x+2)} - \frac{2(x+2)}{x(x+2)} \]

Step 3: Expand and simplify:

\[ \frac{4x - (2x + 4)}{x(x+2)} \] \[ \frac{2x - 4}{x(x+2)} \]

Final Answer:

\[ \frac{2(x-2)}{x(x+2)} \]

Domain Restrictions: \( x \neq 0, -2 \).

Example 2: Adding Rational Expressions with Quadratic Denominators

Given:

\[ \frac{x+5}{x^2-9} + \frac{3}{x-3} \]

Step 1: Factor \( x^2-9 \):

\[ x^2 - 9 = (x-3)(x+3) \]

Step 2: Identify the LCD:

\[ (x-3)(x+3) \]

Step 3: Rewrite both fractions with the LCD:

\[ \frac{x+5}{(x-3)(x+3)} + \frac{3(x+3)}{(x-3)(x+3)} \]

Step 4: Expand the numerators:

\[ \frac{x+5 + 3x+9}{(x-3)(x+3)} \] \[ \frac{4x + 14}{(x-3)(x+3)} \]

Final Answer:

\[ \frac{2(2x+7)}{(x-3)(x+3)} \]

Domain Restrictions: \( x \neq \pm 3 \).

Example 3: Subtracting Rational Expressions with Factored Denominators

Given:

\[ \frac{7}{x^2+2x} - \frac{4}{x} \]

Step 1: Factor \( x^2+2x \):

\[ x^2+2x = x(x+2) \]

Step 2: Identify the LCD:

\[ x(x+2) \]

Step 3: Rewrite both fractions with the LCD:

\[ \frac{7}{x(x+2)} - \frac{4(x+2)}{x(x+2)} \]

Step 4: Expand and simplify:

\[ \frac{7 - 4(x+2)}{x(x+2)} \] \[ \frac{7 - 4x - 8}{x(x+2)} \] \[ \frac{-4x - 1}{x(x+2)} \]

Final Answer:

\[ \frac{-4x-1}{x(x+2)} \]

Domain Restrictions: \( x \neq 0, -2 \).

Multimedia Resources from Media4Math.com

• Review of rational expressions: https://www.media4math.com/library/73874/asset-preview
• Slide Show: Animated Clip Art of Rational Expressions: https://www.media4math.com/library/slideshow/slide-show-animated-clip-art-rational-expressions
• A collection of animated clip art on Rational Expressions: https://www.media4math.com/MathClipArt--AnimatedClipArtRationalExpressions

Quiz

Directions: Add or subtract the following rational expressions and simplify. Show all work.

1. \( \frac{2}{x+1} + \frac{5}{x+1} \) 
   
    
2. \( \frac{4}{x} + \frac{3}{x+3} \) 
   
    
3. \( \frac{x+2}{x^2-1} - \frac{1}{x-1} \) 
   
    
4. \( \frac{5}{x+2} + \frac{3}{x+5} \) 
   
    
5. \( \frac{6}{x^2-4} - \frac{2}{x+2} \) 
   
    
6. Real-world problem: Two workers complete a job at rates of \( \frac{3}{x+1} \) and \( \frac{2}{x} \). Find the combined work rate. 
   
    
7. Find the LCD: \( \frac{1}{x+3} + \frac{1}{x+5} \) 
   
    
8. Simplify: \( \frac{2x+3}{x^2+5x+6} + \frac{3}{x+3} \) 
   
    
9. Find the domain restrictions for: \( \frac{5x+4}{x^2-4} \) 
   
    
10. True or False: The LCD of \( \frac{3}{x+1} + \frac{4}{x+2} \) is \( x+3 \). 
    
     

Answer Key

1. \( \frac{7}{x+1} \)
2. \( \frac{4(x+3)+3x}{x(x+3)} = \frac{4x+12+3x}{x(x+3)} = \frac{7x+12}{x(x+3)} \)
3. \( \frac{x+2}{(x-1)(x+1)} - \frac{1(x+1)}{(x-1)(x+1)} = \frac{x+2 - (x+1)}{(x-1)(x+1)} = \frac{1}{(x-1)(x+1)} \)
4. \( \frac{5(x+5) + 3(x+2)}{(x+2)(x+5)} = \frac{5x+25+3x+6}{(x+2)(x+5)} = \frac{8x+31}{(x+2)(x+5)} \)
5. \( \frac{6}{(x-2)(x+2)} - \frac{2(x-2)}{(x-2)(x+2)} = \frac{6 - 2(x-2)}{(x-2)(x+2)} = \frac{6-2x+4}{(x-2)(x+2)} = \frac{-2x+10}{(x-2)(x+2)} \)
6. \( \frac{3x + 2(x+1)}{x(x+1)} = \frac{3x+2x+2}{x(x+1)} = \frac{5x+2}{x(x+1)} \) (Total work rate)
7. \( (x+3)(x+5) \) (LCD)
8. \( \frac{2x+3+3(x+2)}{(x+2)(x+3)} = \frac{2x+3+3x+6}{(x+2)(x+3)} = \frac{5x+9}{(x+2)(x+3)} \)
9. \( x \neq \pm 2 \) (Denominator restriction)
10. False