🔟Elementary Algebra Unit 8 – Rational Expressions and Equations

Key Concepts

• Rational expressions are fractions with polynomials in the numerator and denominator
• Rational equations are equations that contain rational expressions
• Rational functions are functions that can be written as a ratio of two polynomials
• Domain of a rational expression or function consists of all real numbers except those that make the denominator equal to zero
• Simplifying rational expressions involves factoring the numerator and denominator and canceling common factors
• Operations with rational expressions include addition, subtraction, multiplication, and division
• Solving rational equations involves finding the values of the variable that make the equation true
• Graphing rational functions requires finding the domain, vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercept

Simplifying Rational Expressions

• Factor the numerator and denominator completely
  • Use factoring techniques such as common factors, difference of squares, and trinomial factoring
• Cancel common factors in the numerator and denominator
  • Factors can be canceled if they appear in both the numerator and denominator
• Simplify any remaining terms in the numerator and denominator
• Check the domain of the simplified rational expression
  • The domain consists of all real numbers except those that make the denominator equal to zero
• Example: Simplify $\frac{x^2-4}{x^2-5x+6}$
  • Factor the numerator: $x^2-4 = (x+2)(x-2)$
  • Factor the denominator: $x^2-5x+6 = (x-2)(x-3)$
  • Cancel the common factor $(x-2)$: $\frac{x+2}{x-3}$
  • The domain is all real numbers except $x=3$

Operations with Rational Expressions

• Addition and subtraction require a common denominator
  • Find the least common multiple (LCM) of the denominators
  • Multiply the numerator and denominator of each rational expression by the appropriate factor to obtain the common denominator
  • Add or subtract the numerators and keep the common denominator
• Multiplication involves multiplying the numerators and denominators separately
  • Simplify the resulting rational expression by canceling common factors
• Division of rational expressions is performed by multiplying the first expression by the reciprocal of the second expression
  • Simplify the resulting rational expression by canceling common factors
• Example: Simplify $\frac{2}{x+1} + \frac{3}{x-1}$
  • LCM of the denominators is $(x+1)(x-1)$
  • $\frac{2}{x+1} \cdot \frac{x-1}{x-1} + \frac{3}{x-1} \cdot \frac{x+1}{x+1} = \frac{2(x-1)}{(x+1)(x-1)} + \frac{3(x+1)}{(x+1)(x-1)}$
  • $\frac{2x-2+3x+3}{(x+1)(x-1)} = \frac{5x+1}{x^2-1}$

Solving Rational Equations

• Multiply both sides of the equation by the least common multiple (LCM) of the denominators to clear the fractions
• Simplify the resulting equation
• Solve the equation using appropriate techniques (factoring, quadratic formula, etc.)
• Check the solutions by substituting them back into the original equation
  • Reject any solutions that make the denominators equal to zero
• Example: Solve $\frac{2}{x+1} = \frac{3}{x-1}$
  • Multiply both sides by the LCM $(x+1)(x-1)$: $2(x-1) = 3(x+1)$
  • Simplify: $2x-2 = 3x+3$
  • Solve: $-x = 5$, so $x = -5$
  • Check the solution: $\frac{2}{-5+1} = \frac{3}{-5-1}$ is true, and the denominators are not zero

Graphing Rational Functions

• Find the domain of the rational function
  • The domain consists of all real numbers except those that make the denominator equal to zero
• Find the vertical asymptotes
  • Vertical asymptotes occur at the x-values that make the denominator equal to zero
• Find the horizontal asymptote (if it exists)
  • Compare the degrees of the numerator and denominator
    • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0
    • If the degrees are equal, the horizontal asymptote is y = the leading coefficient of the numerator divided by the leading coefficient of the denominator
    • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote
• Find the x-intercepts (if they exist)
  • Set the numerator equal to zero and solve for x
• Find the y-intercept (if it exists)
  • Substitute x = 0 into the rational function and simplify
• Plot the points and asymptotes, and sketch the graph
• Example: Graph $f(x) = \frac{x+1}{x-2}$
  • Domain: all real numbers except x = 2
  • Vertical asymptote: x = 2
  • Horizontal asymptote: y = 1 (degrees are equal, $\frac{1}{1} = 1$)
  • x-intercept: x = -1
  • y-intercept: $f(0) = \frac{1}{-2} = -\frac{1}{2}$

Applications and Word Problems

• Identify the given information and the unknown quantity
• Set up a rational equation or expression based on the problem statement
• Solve the equation or simplify the expression
• Interpret the solution in the context of the problem
• Example: A group of friends decides to split the cost of a $60 gift. If each person contributes$5 less than the original share, two more people can join the group. How many people were in the group originally?
  • Let x be the number of people in the group originally
  • The original share per person is $\frac{60}{x}$
  • The new share per person is $\frac{60}{x+2} = \frac{60}{x} - 5$
  • Set up the equation: $\frac{60}{x+2} = \frac{60}{x} - 5$
  • Solve the equation: x = 6
  • Interpret the solution: There were 6 people in the group originally

Common Mistakes and How to Avoid Them

• Forgetting to factor the numerator and denominator completely before canceling common factors
  • Always factor the numerator and denominator as much as possible
• Canceling terms instead of factors
  • Only cancel factors that appear in both the numerator and denominator
• Forgetting to find the LCM when adding or subtracting rational expressions
  • Always find the LCM of the denominators and adjust the numerators accordingly
• Not checking the domain of the rational expression or function
  • Always consider the values that make the denominator equal to zero and exclude them from the domain
• Forgetting to check the solutions of rational equations
  • Always substitute the solutions back into the original equation to verify their validity and reject any solutions that make the denominators equal to zero
• Misidentifying the horizontal asymptote
  • Compare the degrees of the numerator and denominator to determine the existence and equation of the horizontal asymptote
• Plotting the vertical asymptote as a vertical line
  • Vertical asymptotes are not part of the graph; they represent the x-values that the graph approaches but never reaches

Practice Problems and Solutions

1. Simplify $\frac{x^2+3x}{x^2-4}$

   • Solution: $\frac{x(x+3)}{(x+2)(x-2)} = \frac{x}{x-2}$, domain: all real numbers except x = 2

2. Perform the operation and simplify: $\frac{3}{x+2} - \frac{1}{x-1}$

   • Solution: $\frac{3(x-1)}{(x+2)(x-1)} - \frac{1(x+2)}{(x+2)(x-1)} = \frac{3x-3-x-2}{(x+2)(x-1)} = \frac{2x-5}{x^2+x-2}$

3. Solve the rational equation: $\frac{2}{x-1} + \frac{3}{x+2} = \frac{5}{x+1}$

   • Solution: x = -5 or x = 4, but x = -2 is rejected because it makes a denominator equal to zero

4. Graph the rational function: $f(x) = \frac{x-1}{x+3}$

   • Solution: Domain: all real numbers except x = -3, Vertical asymptote: x = -3, Horizontal asymptote: y = 1, x-intercept: x = 1, y-intercept: $f(0) = -\frac{1}{3}$

5. A rectangle's length is 3 units more than its width. If the rectangle's area is 10 square units, find its dimensions.

   • Solution: Let the width be x units. Then, the length is x + 3 units. Set up the equation: $x(x+3) = 10$. Solve the equation: x = 1 or x = 2. The dimensions are either 1 unit by 4 units or 2 units by 5 units.