Skip to Content
OpenStax Logo
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index

1.1 Introduction to Whole Numbers

  • Place Value as in Figure 1.3.
  • Name a Whole Number in Words
    1. Step 1. Start at the left and name the number in each period, followed by the period name.
    2. Step 2. Put commas in the number to separate the periods.
    3. Step 3. Do not name the ones period.
  • Write a Whole Number Using Digits
    1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
    2. Step 2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
    3. Step 3. Name the number in each period and place the digits in the correct place value position.
  • Round Whole Numbers
    1. Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
    2. Step 2. Underline the digit to the right of the given place value.
    3. Step 3. Is this digit greater than or equal to 5?
      • Yes—add 1 to the digit in the given place value.
      • No—do not change the digit in the given place value.
    4. Step 4. Replace all digits to the right of the given place value with zeros.
  • Divisibility Tests: A number is divisible by:
    • 2 if the last digit is 0, 2, 4, 6, or 8.
    • 3 if the sum of the digits is divisible by 3.
    • 5 if the last digit is 5 or 0.
    • 6 if it is divisible by both 2 and 3.
    • 10 if it ends with 0.
  • Find the Prime Factorization of a Composite Number
    1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
    2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
    3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
    4. Step 4. Write the composite number as the product of all the circled primes.
  • Find the Least Common Multiple by Listing Multiples
    1. Step 1. List several multiples of each number.
    2. Step 2. Look for the smallest number that appears on both lists.
    3. Step 3. This number is the LCM.
  • Find the Least Common Multiple Using the Prime Factors Method
    1. Step 1. Write each number as a product of primes.
    2. Step 2. List the primes of each number. Match primes vertically when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Multiply the factors.

1.2 Use the Language of Algebra

  • Notation                      The result is…

    a+bthe sum ofaandbabthe difference ofaandba·b,ab,(a)(b)(a)b,a(b)the product ofaandba÷b,a/b,ab,bathe quotient ofaandba+bthe sum ofaandbabthe difference ofaandba·b,ab,(a)(b)(a)b,a(b)the product ofaandba÷b,a/b,ab,bathe quotient ofaandb
  • Inequality

    a<bis readais less thanbais to the left ofbon the number linea>bis readais greater thanbais to the right ofbon the number linea<bis readais less thanbais to the left ofbon the number linea>bis readais greater thanbais to the right ofbon the number line
  • Inequality Symbols                 Words

    abaisnot equal toba<baisless thanbabaisless than or equal toba>baisgreater thanbabaisgreater than or equal tobabaisnot equal toba<baisless thanbabaisless than or equal toba>baisgreater thanbabaisgreater than or equal tob
  • Grouping Symbols
    • Parentheses ()()
    • Brackets [][]
    • Braces {}{}
  • Exponential Notation
    • anan means multiply aa by itself, nn times. The expression anan is read aa to the nthnth power.
  • Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
    1. Step 1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
    2. Step 2. Exponents: Simplify all expressions with exponents.
    3. Step 3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
    4. Step 4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
  • Combine Like Terms
    1. Step 1. Identify like terms.
    2. Step 2. Rearrange the expression so like terms are together.
    3. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

1.3 Add and Subtract Integers

  • Addition of Positive and Negative Integers
    5+35+(−3)8−8both positive,both negative,sum positivesum negative−5+35+(−3)−22different signs,different signs,more negativesmore positivessum negativesum positive5+35+(−3)8−8both positive,both negative,sum positivesum negative−5+35+(−3)−22different signs,different signs,more negativesmore positivessum negativesum positive
  • Property of Absolute Value: |n|0|n|0 for all numbers. Absolute values are always greater than or equal to zero!
  • Subtraction of Integers
    53−5(−3)2−25positives5negativestake away3positivestake away3negatives2 positives2 negatives−535(−3)−885negatives, want to5positives, want tosubtract3positivessubtract3negativesneed neutral pairsneed neutral pairs53−5(−3)2−25positives5negativestake away3positivestake away3negatives2 positives2 negatives−535(−3)−885negatives, want to5positives, want tosubtract3positivessubtract3negativesneed neutral pairsneed neutral pairs
  • Subtraction Property: Subtracting a number is the same as adding its opposite.

1.4 Multiply and Divide Integers

  • Multiplication and Division of Two Signed Numbers
    • Same signs—Product is positive
    • Different signs—Product is negative
  • Strategy for Applications
    1. Step 1. Identify what you are asked to find.
    2. Step 2. Write a phrase that gives the information to find it.
    3. Step 3. Translate the phrase to an expression.
    4. Step 4. Simplify the expression.
    5. Step 5. Answer the question with a complete sentence.

1.5 Visualize Fractions

  • Equivalent Fractions Property: If a,b,ca,b,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.
  • Fraction Division: If a,b,candda,b,candd are numbers where b0,c0,andd0,b0,c0,andd0, then ab÷cd=ab·dc.ab÷cd=ab·dc. To divide fractions, multiply the first fraction by the reciprocal of the second.
  • Fraction Multiplication: If a,b,candda,b,candd are numbers where b0,andd0,b0,andd0, then ab·cd=acbd.ab·cd=acbd. To multiply fractions, multiply the numerators and multiply the denominators.
  • Placement of Negative Sign in a Fraction: For any positive numbers aandb,aandb, ab=ab=ab.ab=ab=ab.
  • Property of One: aa=1;aa=1; Any number, except zero, divided by itself is one.
  • 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.
  • 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.

1.6 Add and Subtract Fractions

  • Fraction Addition and Subtraction: If a,b,andca,b,andc are numbers where c0,c0, then
    ac+bc=a+bcac+bc=a+bc and acbc=abc.acbc=abc.
    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
  • Strategy for Adding or Subtracting 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. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
  • 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.

1.7 Decimals

  • Name a Decimal
    1. Step 1. Name the number to the left of the decimal point.
    2. Step 2. Write ”and” for the decimal point.
    3. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number.
    4. Step 4. Name the decimal place of the last digit.
  • Write a Decimal
    1. Step 1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right.
    2. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
    3. Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
    4. Step 4. Fill in zeros for place holders as needed.
  • Round a Decimal
    1. Step 1. Locate the given place value and mark it with an arrow.
    2. Step 2. Underline the digit to the right of the place value.
    3. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value.
    4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.
  • Add or Subtract Decimals
    1. Step 1. Write the numbers so the decimal points line up vertically.
    2. Step 2. Use zeros as place holders, as needed.
    3. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
  • Multiply Decimals
    1. Step 1. Determine the sign of the product.
    2. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
    3. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors.
    4. Step 4. Write the product with the appropriate sign.
  • Multiply a Decimal by a Power of Ten
    1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
    2. Step 2. Add zeros at the end of the number as needed.
  • Divide Decimals
    1. Step 1. Determine the sign of the quotient.
    2. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places - adding zeros as needed.
    3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
    4. Step 4. Write the quotient with the appropriate sign.
  • Convert a Decimal to a Proper Fraction
    1. Step 1. Determine the place value of the final digit.
    2. Step 2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.
  • Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

1.8 The Real Numbers

  • Square Root Notation
    mm is read ‘the square root of m.’ If m=n2,m=n2, then m=n,m=n, for n0.n0.
  • Order Decimals
    1. Step 1. Write the numbers one under the other, lining up the decimal points.
    2. Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
    3. Step 3. Compare the numbers as if they were whole numbers.
    4. Step 4. Order the numbers using the appropriate inequality sign.

1.9 Properties of Real Numbers

  • Commutative Property of
    • Addition: If a,ba,b are real numbers, then a+b=b+a.a+b=b+a.
    • Multiplication: If a,ba,b are real numbers, then a·b=b·a.a·b=b·a. When adding or multiplying, changing the order gives the same result.
  • Associative Property of
    • Addition: If a,b,ca,b,c are real numbers, then (a+b)+c=a+(b+c).(a+b)+c=a+(b+c).
    • Multiplication: If a,b,ca,b,c are real numbers, then (a·b)·c=a·(b·c).(a·b)·c=a·(b·c).
      When adding or multiplying, changing the grouping gives the same result.
  • Distributive Property: If a,b,ca,b,c are real numbers, then
    • a(b+c)=ab+aca(b+c)=ab+ac
    • (b+c)a=ba+ca(b+c)a=ba+ca
    • a(bc)=abaca(bc)=abac
    • (bc)a=baca(bc)a=baca
  • Identity Property
    • of Addition: For any real number a:a+0=a0+a=aa:a+0=a0+a=a
      0 is the additive identity
    • of Multiplication: For any real number a:a·1=a1·a=aa:a·1=a1·a=a
      11 is the multiplicative identity
  • Inverse Property
    • of Addition: For any real number a,a+(a)=0.a,a+(a)=0. A number and its opposite add to zero. aa is the additive inverse of a.a.
    • of Multiplication: For any real number a,(a0)a·1a=1.a,(a0)a·1a=1. A number and its reciprocal multiply to one. 1a1a is the multiplicative inverse of a.a.
  • Properties of Zero
    • For any real number a,a,
      a·0=00·a=0a·0=00·a=0 – The product of any real number and 0 is 0.
    • 0a=00a=0 for a0a0 – Zero divided by any real number except zero is zero.
    • a0a0 is undefined – Division by zero is undefined.

1.10 Systems of Measurement

  • Metric System of Measurement
    • Length
      1 kilometer (km)=1,000 m 1 hectometer (hm)=100 m 1 dekameter (dam)=10 m 1 meter (m)=1 m 1 decimeter (dm)=0.1 m 1 centimeter (cm)=0.01 m 1 millimeter (mm)=0.001 m 1 meter=100 centimeters 1 meter=1,000 millimeters1 kilometer (km)=1,000 m 1 hectometer (hm)=100 m 1 dekameter (dam)=10 m 1 meter (m)=1 m 1 decimeter (dm)=0.1 m 1 centimeter (cm)=0.01 m 1 millimeter (mm)=0.001 m 1 meter=100 centimeters 1 meter=1,000 millimeters
    • Mass
      1 kilogram (kg)=1,000 g 1 hectogram (hg)=100 g 1 dekagram (dag)=10 g 1 gram (g)=1 g 1 decigram (dg)=0.1 g 1 centigram (cg)=0.01 g 1 milligram (mg)=0.001 g 1 gram=100 centigrams 1 gram=1,000 milligrams1 kilogram (kg)=1,000 g 1 hectogram (hg)=100 g 1 dekagram (dag)=10 g 1 gram (g)=1 g 1 decigram (dg)=0.1 g 1 centigram (cg)=0.01 g 1 milligram (mg)=0.001 g 1 gram=100 centigrams 1 gram=1,000 milligrams
    • Capacity
      1 kiloliter (kL)=1,000 L 1 hectoliter (hL)=100 L 1 dekaliter (daL)=10 L 1 liter (L)=1 L 1 deciliter (dL)=0.1 L 1 centiliter (cL)=0.01 L 1 milliliter (mL)=0.001 L 1 liter=100 centiliters 1 liter=1,000 milliliters1 kiloliter (kL)=1,000 L 1 hectoliter (hL)=100 L 1 dekaliter (daL)=10 L 1 liter (L)=1 L 1 deciliter (dL)=0.1 L 1 centiliter (cL)=0.01 L 1 milliliter (mL)=0.001 L 1 liter=100 centiliters 1 liter=1,000 milliliters
  • Temperature Conversion
    • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula C=59(F32)C=59(F32)
    • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula F=95C+32F=95C+32
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
Citation information

© Sep 2, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.