Elementary Algebra 2e

Chapter 1 Foundations
15 Topics
Introduction
1.1 Introduction to Whole Numbers
1.2 Use the Language of Algebra
1.3 Add and Subtract Integers
1.4 Multiply and Divide Integers
1.5 Visualize Fractions
1.6 Add and Subtract Fractions
1.7 Decimals
1.8 The Real Numbers
1.9 Properties of Real Numbers
1.10 Systems of Measurement
Chapter Review Key Terms
Key Concepts
Review Exercises
Practice Test
Chapter 2 Solving Linear Equations and Inequalities
12 Topics
Introduction
2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
2.2 Solve Equations using the Division and Multiplication Properties of Equality
2.3 Solve Equations with Variables and Constants on Both Sides
2.4 Use a General Strategy to Solve Linear Equations
2.5 Solve Equations with Fractions or Decimals
2.6 Solve a Formula for a Specific Variable
2.7 Solve Linear Inequalities
Key Terms
Key Concepts
Review Exercises
Practice Test
Chapter 3 Math Models
11 Topics
Introduction
3.1 Use a Problem-Solving Strategy
3.2 Solve Percent Applications
3.3 Solve Mixture Applications
3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
3.5 Solve Uniform Motion Applications
3.6 Solve Applications with Linear Inequalities
Key Terms
Key Concepts
Review Exercises
Practice Test
Chapter 4 Graphs
12 Topics
Introduction
4.1 Use the Rectangular Coordinate System
4.2 Graph Linear Equations in Two Variables
4.3 Graph with Intercepts
4.4 Understand Slope of a Line
4.5 Use the Slope-Intercept Form of an Equation of a Line
4.6 Find the Equation of a Line
4.7 Graphs of Linear Inequalities
Chapter Review
Key Terms
Review Exercises
Practice Test
Chapter 5 Systems of Linear Equations
5 Topics
Introduction
5.1 Solve Systems of Equations by Graphing
5.3 Solve Systems of Equations by Elimination
5.5 Solve Mixture Applications with Systems of Equations
5.4 Solve Applications with Systems of Equations
Chapter 6 Polynomials
12 Topics
Introduction
6.1 Add and Subtract Polynomials
6.2 Use Multiplication Properties of Exponents
6.3 Multiply Polynomials
6.4 Special Products
6.5 Divide Monomials
6.6 Divide Polynomials
6.7 Integer Exponents and Scientific Notation
Key Terms
Key Concepts
Review Exercises
Practice Test
7 Factoring
11 Topics
Introduction
7.1 Greatest Common Factor and Factor by Grouping
7.2 Factor Trinomials of the Form x2+bx+c
7.3 Factor Trinomials of the Form ax2+bx+c
7.4 Factor Special Products
7.5 General Strategy for Factoring Polynomials
7.6 Quadratic Equations
Key Terms
Key Concepts
Review Exercises
Practice Test
8 Rational Expressions and Equations
14 Topics
Introduction
8.1 Simplify Rational Expressions
8.2 Multiply and Divide Rational Expressions
8.3 Add and Subtract Rational Expressions with a Common Denominator
8.4 Add and Subtract Rational Expressions with Unlike Denominators
8.5 Simplify Complex Rational Expressions
8.6 Solve Rational Equations
8.7 Solve Proportion and Similar Figure Applications
8.8 Solve Uniform Motion and Work Applications
8.9 Use Direct and Inverse Variation
Key Terms
Key Concepts
Review Exercises
Practice Test
9 Roots and Radicals
13 Topics
Introduction
9.1 Simplify and Use Square Roots
9.2 Simplify Square Roots
9.3 Add and Subtract Square Roots
9.4 Multiply Square Roots
9.5 Divide Square Roots
9.6 Solve Equations with Square Roots
9.7 Higher Roots
9.8 Rational Exponents
Key Terms
Key Concepts
Review Exercises
Practice Test
10 Quadratic Equations
10 Topics
Introduction
10.1 Solve Quadratic Equations Using the Square Root Property
10.2 Solve Quadratic Equations by Completing the Square
10.3 Solve Quadratic Equations Using the Quadratic Formula
10.4 Solve Applications Modeled by Quadratic Equations
10.5 Graphing Quadratic Equations in Two Variables
Key Terms
Key Concepts
Review Exercises
Practice Test
Answer Key
1 Topic
Index
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8.5 Simplify Complex Rational Expressions

Elementary Algebra 2e 8 Rational Expressions and Equations 8.5 Simplify Complex Rational Expressions

Learning Objectives

By the end of this section, you will be able to:

  • Simplify a complex rational expression by writing it as division
  • Simplify a complex rational expression by using the LCD
Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:

In this section we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Complex Rational Expression

A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression.

Here are a few complex rational expressions:

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.

We noted that fraction bars tell us to divide, so rewrote it as the division problem

Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify rational expressions. We write it as if we were dividing two fractions.

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

How To

Simplify a complex rational expression by writing it as division.

  • Step 1. Simplify the numerator and denominator.
  • Step 2. Rewrite the complex rational expression as a division problem.
  • Step 3. Divide the expressions.

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 8.51. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by LCDLCD we are multiplying by 1, so the value stays the same.

How To

Simplify a complex rational expression by using the LCD.

  • Step 1. Find the LCD of all fractions in the complex rational expression.
  • Step 2. Multiply the numerator and denominator by the LCD.
  • Step 3. Simplify the expression.

Be sure to start by factoring all the denominators so you can find the LCD.

Section 8.5 Exercises

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Simplify

In the following exercises, use either method.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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