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
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1.5 Visualize Fractions

Elementary Algebra 2e Chapter 1 Foundations 1.5 Visualize Fractions

Learning Objectives

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

  • Find equivalent fractions
  • Simplify fractions
  • Multiply fractions
  • Divide fractions
  • Simplify expressions written with a fraction bar
  • Translate phrases to expressions with fractions

Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction 1313 means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See Figure 1.11. The fraction 2323 represents two of three equal parts. In the fraction 23,23, the 2 is called the numerator and the 3 is called the denominator.

Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.

Figure 1.11 The circle on the left has been divided into 3 equal parts. Each part is 1313 of the 3 equal parts. In the circle on the right, 2323 of the circle is shaded (2 of the 3 equal parts).

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

  • 2/3 is simplified because there are no common factors of 2 and 3.
  • 10/15 is not simplified because 55 is a common factor of 10 and 15.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with 34.

A rectangle made up of four squares in a row. The first three squares are shaded.

Now we’ll take 1/2 of 3/4.

A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.

Notice that now, the whole is divided into 8 equal parts. So 

To multiply fractions, we multiply the numerators and multiply the denominators.

Section 1.5 Exercises

Practice Makes Perfect

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.A table is shown that is made up of four columns and seven rows. The first row reads “I can…” in the first column, “Confidently” in the second column, “With some help” in the third column and “No – I don’t get it” in the last column. The next row down in the first column reads “find equivalent fractions”, under this reads “simplify fractions”, under this reads “multiply fractions”, under this reads “divide fractions”, under this reads “Simplify expressions written with a fraction bar” and under this reads “translate phrases to expressions with fractions.”

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

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