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  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify fractions
  • Multiply and divide fractions
  • Add and subtract fractions
  • Use the order of operations to simplify fractions
  • Evaluate variable expressions with fractions
Be Prepared 1.3

A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra 2e chapter, Foundations.

Simplify Fractions

A fraction is a way to represent parts of a whole. The fraction 2323 represents two of three equal parts. See Figure 1.5. In the fraction 23,23, the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.

Figure shows a circle divided in three equal parts. 2 of these are shaded.
Figure 1.5 In the circle, 2323 of the circle is shaded—2 of the 3 equal parts.

Fraction

A fraction is written ab,ab, where b0b0 and

a is the numerator and b is the denominator.

A fraction represents parts of a whole. The denominator bb is the number of equal parts the whole has been divided into, and the numerator aa indicates how many parts are included.

Fractions that have the same value are equivalent fractions. The Equivalent Fractions

Property allows us to find equivalent fractions and also simplify fractions.

Equivalent Fractions Property

If a, b, and c are numbers where b0,c0,b0,c0,

then ab=a·cb·cab=a·cb·c and a·cb·c=ab.a·cb·c=ab.

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

For example,

  2323 is simplified because there are no common factors of 2 and 3.3.

  10151015 is not simplified because 5 is a common factor of 10 and 15.15.

We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.

Example 1.24 How To Simplify a Fraction

Simplify: 315770.315770.

Try It 1.47

Simplify: 69120.69120.

Try It 1.48

Simplify: 120192.120192.

We now summarize the steps you should follow to simplify fractions.

How To

Simplify a fraction.

  1. Step 1. Rewrite the numerator and denominator to show the common factors.
    If needed, factor the numerator and denominator into prime numbers first.
  2. Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
  3. Step 3. Multiply any remaining factors.

Multiply and Divide Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions.

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

Fraction Multiplication

If a, b, c, and d are numbers where b0,b0, and d0,d0, then

ab·cd=acbdab·cd=acbd

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

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1.25, we will multiply a negative by a negative, so the product will be positive.

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as a1.a1. So, for example, 3=31.3=31.

Example 1.25

Multiply: 125(−20x).125(−20x).

Try It 1.49

Multiply: 113(−9a).113(−9a).

Try It 1.50

Multiply: 137(−14b).137(−14b).

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of 2323 is 32.32. Since 4 is written in fraction form as 41,41, the reciprocal of 4 is 14.14.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction Division

If a, b, c, and d are numbers where b0,c0,b0,c0, and d0,d0, then

ab÷cd=ab·dcab÷cd=ab·dc

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say b0,b0, c0,c0, and d0,d0, to be sure we don’t divide by zero!

Example 1.26

Find the quotient: 718÷(1427).718÷(1427).

Try It 1.51

Divide: 727÷(3536).727÷(3536).

Try It 1.52

Divide: 514÷(1528).514÷(1528).

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Complex Fraction

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

Some examples of complex fractions are:

6733458x2566733458x256

To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction 34583458 means 34÷58.34÷58.

Example 1.27

Simplify: x2xy6.x2xy6.

Try It 1.53

Simplify: a8ab6.a8ab6.

Try It 1.54

Simplify: p2pq8.p2pq8.

Add and Subtract Fractions

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Fraction Addition and Subtraction

If a, b, and c are numbers where c0,c0, then

ac+bc=a+bcandacbc=abcac+bc=a+bcandacbc=abc

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

Example 1.28 How to Add or Subtract Fractions

Add: 712+518.712+518.

Try It 1.55

Add: 712+1115.712+1115.

Try It 1.56

Add: 1315+1720.1315+1720.

How To

Add or subtract fractions.

  1. Step 1. Do they have a common denominator?
    • Yes—go to step 2.
    • No—rewrite each fraction with the LCD (least common denominator).
      • Find the LCD.
      • Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Step 2. Add or subtract the fractions.
  3. Step 3. Simplify, if possible.

We now have all four operations for fractions. Table 1.3 summarizes fraction operations.

Fraction Multiplication Fraction Division
ab·cd=acbdab·cd=acbd ab÷cd=ab·dcab÷cd=ab·dc
Multiply the numerators and multiply the denominators Multiply the first fraction by the reciprocal of the second.
Fraction Addition Fraction Subtraction
ac+bc=a+bcac+bc=a+bc acbc=abcacbc=abc
Add the numerators and place the sum over the common denominator. Subtract the numerators and place the difference over the common denominator.
To multiply or divide fractions, an LCD is NOT needed.
To add or subtract fractions, an LCD is needed.
Table 1.3

When starting an exercise, always identify the operation and then recall the methods needed for that operation.

Example 1.29

Simplify: 5x63105x6310 5x6·310.5x6·310.

Try It 1.57

Simplify: 3a4893a489 3a4·89.3a4·89.

Try It 1.58

Simplify: 4k5164k516 4k5·16.4k5·16.

Use the Order of Operations to Simplify Fractions

The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

How To

Simplify an expression with a fraction bar.

  1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Step 2. Simplify the fraction.

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

−13=13negativepositive=negative−13=13negativepositive=negative
1−3=13positivenegative=negative1−3=13positivenegative=negative

Placement of Negative Sign in a Fraction

For any positive numbers a and b,

ab=ab=abab=ab=ab

Example 1.30

Simplify: 4(3)+6(2)3(2)2.4(3)+6(2)3(2)2.

Try It 1.59

Simplify: 8(2)+4(3)5(2)+3.8(2)+4(3)5(2)+3.

Try It 1.60

Simplify: 7(1)+9(3)5(3)2.7(1)+9(3)5(3)2.

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.

Example 1.31 How to Simplify Complex Fractions

Simplify: (12)24+32.(12)24+32.

Try It 1.61

Simplify: ( 1 3)223+2.( 1 3)223+2.

Try It 1.62

Simplify: 1+42(14)2.1+42(14)2.

How To

Simplify complex fractions.

  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Divide the numerator by the denominator. Simplify if possible.

Example 1.32

Simplify: 12+233416.12+233416.

Try It 1.63

Simplify: 13+123413.13+123413.

Try It 1.64

Simplify: 231214+13.231214+13.

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Example 1.33

Evaluate 2x2y2x2y when x=14x=14 and y=23.y=23.

Try It 1.65

Evaluate 3ab23ab2 when a=23a=23 and b=12.b=12.

Try It 1.66

Evaluate 4c3d4c3d when c=12c=12 and d=43.d=43.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with fractions.

Section 1.3 Exercises

Practice Makes Perfect

Simplify Fractions

In the following exercises, simplify.

143.

1086310863

144.

1044810448

145.

120252120252

146.

182294182294

147.

14x221y14x221y

148.

24a32b224a32b2

149.

210a2110b2210a2110b2

150.

30x2105y230x2105y2

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

151.

34(49)34(49)

152.

38·41538·415

153.

(1415)(920)(1415)(920)

154.

(910)(2533)(910)(2533)

155.

(6384)(4490)(6384)(4490)

156.

(3360)(4088)(3360)(4088)

157.

37·21n37·21n

158.

56·30m56·30m

159.

34÷x1134÷x11

160.

25÷y925÷y9

161.

518÷(1524)518÷(1524)

162.

718÷(1427)718÷(1427)

163.

8u15÷12v258u15÷12v25

164.

12r25÷18s3512r25÷18s35

165.

34÷(−12)34÷(−12)

166.

−15÷(53)−15÷(53)

In the following exercises, simplify.

167.

82112358211235

168.

91633409163340

169.

452452

170.

53105310

171.

m3n2m3n2

172.

38y1238y12

Add and Subtract Fractions

In the following exercises, add or subtract.

173.

712+58712+58

174.

512+38512+38

175.

712916712916

176.

716512716512

177.

1330+25421330+2542

178.

2330+5482330+548

179.

3956223539562235

180.

3349183533491835

181.

23(34)23(34)

182.

34(45)34(45)

183.

x3+14x3+14

184.

x514x514

185.


23+1623+16
23÷1623÷16

186.


25182518
25·1825·18

187.


5n6÷8155n6÷815
5n68155n6815

188.


3a8÷7123a8÷712
3a87123a8712

189.


4x9564x956
4k9·564k9·56

190.


3y8433y843
3y8·433y8·43

191.


5a3+(106)5a3+(106)
5a3÷(106)5a3÷(106)

192.


2b5+8152b5+815
2b5÷8152b5÷815

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

193.

5·63·44·52·35·63·44·52·3

194.

8·97·65·69·28·97·65·69·2

195.

523235523235

196.

624246624246

197.

7·42(85)9·33·57·42(85)9·33·5

198.

9·73(128)8·76·69·73(128)8·76·6

199.

9(82)3(157)6(71)3(179)9(82)3(157)6(71)3(179)

200.

8(92)4(149)7(83)3(169)8(92)4(149)7(83)3(169)

201.

23+42(23)223+42(23)2

202.

3332(34)23332(34)2

203.

(35)2(37)2(35)2(37)2

204.

(34)2(58)2(34)2(58)2

205.

213+15213+15

206.

514+13514+13

207.

782312+38782312+38

208.

343514+25343514+25

Mixed Practice

In the following exercises, simplify.

209.

38÷(310)38÷(310)

210.

312÷(59)312÷(59)

211.

38+51238+512

212.

18+71218+712

213.

715y4715y4

214.

38x1138x11

215.

1112a·9a161112a·9a16

216.

10y13·815y10y13·815y

217.

12+23·51212+23·512

218.

13+25·3413+25·34

219.

135÷110135÷110

220.

156÷112156÷112

221.

3816+343816+34

222.

25+583425+5834

223.

12(920415)12(920415)

224.

8(151656)8(151656)

225.

58+16192458+161924

226.

16+310143016+3101430

227.

(59+16)÷(2312)(59+16)÷(2312)

228.

(34+16)÷(5813)(34+16)÷(5813)

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

229.

710w710w when
w=12w=12 w=12w=12

230.

512w512w when
w=14w=14 w=14w=14

231.

2x2y32x2y3 when
x=23x=23 and y=12y=12

232.

8u2v38u2v3 when
u=34u=34 and v=12v=12

233.

a+baba+bab when
a=−3,b=8a=−3,b=8

234.

rsr+srsr+s when
r=10,s=−5r=10,s=−5

Writing Exercises

235.

Why do you need a common denominator to add or subtract fractions? Explain.

236.

How do you find the LCD of 2 fractions?

237.

Explain how you find the reciprocal of a fraction.

238.

Explain how you find the reciprocal of a negative number.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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