Book cover for Beginning Algebra

Beginning Algebra

John Hornsby; Terry McGinnis; Margaret L. Lial

ISBN #9780136881025

13th Edition

5,472 Questions

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22,138 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter extensively covers rational expressions and their applications. Students learn how to simplify rational expressions by factoring and canceling common factors, perform arithmetic operations using the LCD, and simplify complex fractions using two different methods. The chapter also addresses solving equations involving rational expressions with a focus on identifying extraneous solutions. Real-world applications include work rate problems, distance–rate–time problems, and variations (both direct and inverse), connecting abstract algebraic manipulation to practical scenarios.

Learning Objectives

1

Simplify rational expressions by factoring completely and applying the fundamental property to cancel common factors.

2

Perform operations with rational expressions including multiplication, division, addition, and subtraction using the least common denominator (LCD).

3

Simplify complex fractions using both the division?method and the LCD?multiplication method.

4

Solve equations that involve rational expressions and check for extraneous solutions.

5

Apply rational expressions and variation (direct and inverse) concepts to model and solve real?world problems such as work, distance–rate–time, and proportional relationships.

Key Concepts

CONCEPT

DEFINITION

Rational Expression

An expression that is the quotient of two polynomials where the denominator is not zero.

Lowest Terms

A rational expression is in its lowest terms when the greatest common factor of the numerator and denominator is 1.

Fundamental Property of Rational Expressions

If a nonzero polynomial K is multiplied to both the numerator and denominator of a rational expression, the value of the expression remains unchanged.

Least Common Denominator (LCD)

The smallest expression that contains all the factors of the denominators of given rational expressions, used to combine them through addition or subtraction.

Complex Fraction

A fraction in which the numerator, denominator, or both are themselves fractions.

Direct Variation

A relationship where one variable is a constant multiple of another; stated as y = kx (or y = kx^n for a power variation), where k is the constant of variation.

Inverse Variation

A relationship where one variable is inversely proportional to another; expressed as y = k/x (or y = k/x^n), meaning as one variable increases, the other decreases.

Extraneous Solution

A solution derived algebraically from an equation that does not satisfy the original equation because it makes one or more denominators equal to zero.

Example Problems

Example 1

Fill in each blank with the correct response: The rational expression $\frac{x+5}{x-3}$ is undefined when $x$ is ____ , so $x \neq $ ____. This rational expression is equal to 0 when $x=$ ____

Example 2

Which one of the following rational expressions can be simplified? A. $\frac{x^{2}+2}{x^{2}}$ B. $\frac{x^{2}+2}{2}$ C. $\frac{x^{2}+y^{2}}{y^{2}}$ D. $\frac{x^{2}-5 x}{x}$

Example 3

Which two of the following rational expressions equal $-1 ?$ A. $\frac{2 x+3}{2 x-3}$ B. $\frac{2 x-3}{3-2 x}$ C. $\frac{2 x+3}{3+2 x}$ D. $\frac{2 x+3}{-2 x-3}$

Example 4

Make the correct choice: $\frac{4-r^{2}}{4+r^{2}}(i s /$ is not $)$ equal to -1 .

Example 5

Which one of the following rational expressions is not equivalent to $\frac{x-3}{4-x} ?$ A. $\frac{3-x}{x-4}$ B. $\frac{x+3}{4+x}$ C. $-\frac{3-x}{4-x}$ D. $-\frac{x-3}{x-4}$
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Step-by-Step Explanations

QUESTION

Simplify the rational expression (4x^2 - 16)/(2x^2 - 8x).

STEP-BY-STEP ANSWER:

Step 1: Factor the numerator and denominator completely. The numerator factors as 4(x^2 - 4) = 4(x - 2)(x + 2). The denominator factors as 2x(x - 4) (note: factor common term: 2x^2 - 8x = 2x(x - 4)).
Step 2: Identify any common factors. In this case, there are no common binomial factors since (x - 2) is not the same as (x - 4).
Step 3: Write the simplified expression in lowest terms. Since no common factors cancel (assuming no further factorization possible), the expression remains as (4(x - 2)(x + 2))/(2x(x - 4)).
Step 4: Optionally, simplify the coefficient: 4/2 = 2, so the expression becomes [2(x - 2)(x + 2)]/[x(x - 4)], with the restriction that x ≠ 0 and x ≠ 4.
Final Answer:

Simplifying a Rational Expression

QUESTION

Solve the rational equation (3/(x-2)) + (2/(x+3)) = 5/(x-2).

STEP-BY-STEP ANSWER:

Step 1: Identify the LCD of the denominators. Here the denominators are x-2 and x+3, so the LCD is (x-2)(x+3).
Step 2: Multiply both sides of the equation by the LCD to eliminate fractions:
Step 3: Expand and simplify: 3x + 9 + 2x - 4 = 5x + 15.
Step 4: Combine like terms: (3x + 2x) + (9 - 4) = 5x + 15 gives 5x + 5 = 5x + 15.
Step 5: Subtract 5x from both sides to get 5 = 15, which is a contradiction.
Step 6: Since the equation leads to a false statement, there is no solution (or the solution set is empty). Make sure to check that none of the eliminated potential solutions would have caused division by zero.
Final Answer: No solution.

Solving a Rational Equation

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Common Mistakes

  • Cancelling terms instead of factors: Students sometimes cancel addition terms rather than factors.
  • Failing to factor completely, which leads to missed common factors and an unsimplified expression.
  • Neglecting to identify and exclude values that make any original denominator equal to zero.
  • Improper handling of negative signs, especially when factors are opposites.
  • Incorrectly finding the LCD by summing exponents or over-counting factors rather than taking the maximum occurrence.