Book cover for Intermediate Algebra

Intermediate Algebra

Julie Miller, Molly O'Neill, Nancy Hyde

ISBN #9781259610233

5th Edition

5,614 Questions

Group icon
150,127 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

In this section, learners explored the structure and manipulation of rational expressions and rational equations. Key skills include simplifying expressions by factoring and canceling common factors, solving equations by clearing denominators while respecting domain restrictions, and ensuring that potential extraneous solutions are evaluated. Mastery of these topics is essential for solving more complex algebraic problems and applying these techniques in real-life scenarios.

Learning Objectives

1

Understand the definition and structure of rational expressions and rational equations.

2

Simplify rational expressions by factoring and canceling common factors.

3

Solve rational equations by clearing denominators and identifying extraneous solutions.

4

Recognize and apply domain restrictions in rational expressions and equations.

Key Concepts

CONCEPT

DEFINITION

Rational Expression

An expression that can be written as the quotient of two polynomials, where the denominator is not equal to zero.

Rational Equation

An equation that contains one or more rational expressions, set equal to another expression or number.

Domain Restrictions

The set of all values that the variable cannot assume because they would make any denominator zero.

Simplification

The process of factoring polynomials in the numerator and denominator and canceling common factors to reduce a rational expression to its simplest form.

Clearing Denominators

A method used in solving rational equations by multiplying every term by the least common denominator (LCD) to eliminate fractions.

Example Problems

Example 1

a. A _____ function is a function that can be written as $f ( x ) = \frac { p ( x ) } { a ( x ) } ,$ where $p ( x )$ and $q ( x )$ are polynomial functions and $q ( x ) \neq 0$. b. The domain of a rational function is all real numbers excluding the values of the variable that make the equal to zero. c. For polynomials $p , q ,$ and $r ,$ where $( q \neq 0 \text { and } r \neq 0 ) , \frac { p r } { q r }$ simplifies to _____. d. The ratio $\frac { a - b } { a - b } =$ _____, whereas the ratio $\frac { a - b } { b - a } =$ _____ provided that $a \neq b$.

Example 2

For Exercises 2–6, determine the function values, if possible. (See Example 1.) $$f ( x ) = \frac { 5 } { x + 1 } ; f ( 0 ) , f ( 2 ) , f ( - 1 ) , f ( - 6 )$$

Example 3

Determine the function values, if possible. (See Example 1.) $$k ( x ) = \frac { - 3 } { x + 4 } ; k ( 0 ) , k ( - 1 ) , k ( 2 ) , k ( - 4 )$$

Example 4

Determine the function values, if possible. (See Example 1.) $$m ( x ) = \frac { x - 4 } { x + 6 } ; m ( - 6 ) , m ( - 4 ) , m ( 0 ) , m ( 4 )$$

Example 5

Determine the function values, if possible. (See Example 1.) $$n ( a ) = \frac { 3 a + 1 } { a ^ { 2 } + 1 } ; n ( 1 ) , n ( 0 ) , n \left( - \frac { 1 } { 3 } \right) , n ( - 1 )$$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

How do you simplify the expression (x² - 9) / (x² - x - 12)?

STEP-BY-STEP ANSWER:

Step 1: Factor the numerator. Notice that x² - 9 is a difference of squares and factors as (x - 3)(x + 3).
Step 2: Factor the denominator. Find factors for x² - x - 12 which factors as (x - 4)(x + 3).
Step 3: Identify the common factor (x + 3) in the numerator and the denominator, and cancel it, but note that x must not equal -3, since that would make the original denominator zero.
Final Answer: The simplified expression is (x - 3) / (x - 4) with the restriction that x ≠ -3 and x ≠ 4.

Simplifying a Rational Expression

QUESTION

How do you solve the rational equation 1/(x + 2) = 3/(x - 1)?

STEP-BY-STEP ANSWER:

Step 1: Determine the restrictions: set x + 2 ≠ 0 (x ≠ -2) and x - 1 ≠ 0 (x ≠ 1).
Step 2: Identify the least common denominator (LCD), which is (x + 2)(x - 1).
Step 3: Multiply both sides of the equation by the LCD to eliminate the fractions: (x + 2)(x - 1) * (1/(x + 2)) = (x + 2)(x - 1) * (3/(x - 1)).
Step 4: Simplify both sides; on the left, (x + 2) cancels out, leaving (x - 1), and on the right, (x - 1) cancels out, leaving 3(x + 2).
Step 5: Set up the simplified equation: x - 1 = 3(x + 2).
Step 6: Solve the equation by expanding and rearranging: x - 1 = 3x + 6 leads to -1 - 6 = 3x - x, so -7 = 2x.
Step 7: Divide both sides by 2: x = -7/2.
Step 8: Verify that the solution does not conflict with the restrictions (x ≠ -2 and x ≠ 1).
Final Answer: The valid solution is x = -7/2.

Solving a Rational Equation

Scroll left
Scroll right

Common Mistakes

  • Overlooking domain restrictions by not identifying values that make the denominator zero.
  • Incorrectly factoring expressions, leading to improper cancellation of terms.
  • Cancelling terms that are not common factors, such as parts of a sum rather than the whole factors.
  • Failing to check for extraneous solutions after solving rational equations.