Elementary algebra

OpenStax

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0.1 Pedagogical features

0.3 Arithmetic review

0.4 Basic properties of real numbers

0.5 Basic operations with real numbers

0.6 Algebraic expressions and equations

0.7 Solving linear equations and inequalities

0.8 Factoring polynomials

0.9 Graphing linear equations and inequalities in one and two variables

0.10 Rational expressions

0.11 Roots, radicals, and square root equations

0.12 Quadratic equations

0.13 Systems of linear equations

1 Acknowledgments

Arithmetic review

1 Factors, products, and exponents

1.3 Exponential notation

1.4 Sample set a

2 Prime factorization

2.2 Prime and composite numbers

2.3 The fundamental principle of arithmetic

2.4 Sample set a

2.5 The prime factorization of a whole number

2.6 Sample set b

3 The least common multiple

3.3 Common multiples

3.4 Sample set a

3.5 The least common multiple (lcm)

3.6 Finding the least common multiple

3.7 Sample set b

4 Equivalent fractions

4.2 Equivalent fractions

4.3 Reducing fractions to lowest terms

4.4 Sample set a

4.5 Raising a fraction to higher terms

4.6 Sample set b

5 Operations with fractions

5.2 Multiplication of fractions

5.3 Sample set a

5.4 Division of fractions

5.5 Sample set b

5.6 Addition and subtraction of fractions

5.7 Sample set c

5.8 Sample set d

6 Decimal fractions

6.2 Decimal fractions

6.3 Sample set a

6.4 Adding and subtracting decimal fractions

6.5 Sample set b

6.6 Multiplying decimal fractions

6.7 Sample set c

6.8 Dividing decimal fractions

6.9 Sample set d

6.10 Converting decimal fractions to fractions

6.11 Sample set e

6.12 Converting fractions to decimal fractions

6.13 Sample set f

7.2 The meaning of percent

7.3 Converting a fraction to a percent

7.4 Sample set a

7.5 Converting a decimal to a percent

7.6 Sample set b

7.7 Converting a percent to a decimal

7.8 Sample set c

Basic properties of real numbers

1 Symbols and notations

1.2 Variables and constants

1.3 Symbols of operation, equality, and inequality

1.4 Sample set a

1.5 Practice set a

1.6 Grouping symbols

1.7 Sample set b

1.8 Practice set b

1.9 The order of operations

1.10 Sample set c

1.11 Practice set c

2 The real number line and the real numbers

2.2 The real number line

2.3 The real numbers

2.4 Sample set a

2.5 Practice set a

2.6 Ordering the real numbers

2.7 Sample set b

2.8 Practice set b

2.9 Sample set c

2.10 Practice set c

2.12 Exercises for review

3 Properties of the real numbers

3.2 The closure properties

3.3 The commutative properties

3.4 Sample set a

3.5 Practice set a

3.6 The associative properties

3.7 Sample set b

3.8 Practice set b

3.9 Sample set c

3.10 Practice set c

3.11 The distributive properties

3.12 Sample set d

3.13 Practice set d

3.14 The identity properties

3.15 The inverse properties

3.17 Exercises for review

4.2 Exponential notation

4.3 Sample set a

4.4 Practice set a

4.5 Sample set b

4.6 Practice set b

4.7 Sample set c

4.8 Practice set c

4.9 Reading exponential notation

4.10 The order of operations

4.11 Sample set d

4.12 Practice set d

4.14 Exercises for review

5 Rules of exponents

5.2 The product rule for exponents

5.3 Sample set a

5.4 Practice set a

5.5 Sample set b

5.6 Practice set b

5.7 The quotient rule for exponents

5.8 Sample set c

5.9 Practice set c

5.10 Zero as an exponent

5.11 Sample set d

5.12 Practice set d

5.14 Exercises for review

6 The power rules for exponents

6.2 The power rule for powers

6.3 Sample set a

6.4 Practice set a

6.5 The power rule for products

6.6 Sample set b

6.7 Practice set b

6.8 The power rule for quotients

6.9 Sample set c

6.10 Practice set c

6.12 Exercises for review

7 Summary of key concepts

7.1 Summary of key concepts

8 Exercise supplement

8.1 Exercise supplement

8.2 Symbols and notations ( )

8.3 The real number line and the real numbers ( )

8.4 Properties of the real numbers ( )

8.5 Exponents ( )

8.6 Rules of exponents ( ) - the power rules for exponents ( )

9 Proficiency exam

9.1 Proficiency exam

Basic operations with real numbers

1 Signed numbers

1.2 Positive and negative numbers

1.3 Sample set a

1.4 Practice set a

1.6 Sample set b

1.7 Practice set b

1.9 Exercises for review

2 Absolute value

2.2 Geometric definition of absolute value

2.3 Algebraic definition of absolute value

2.4 Sample set a

2.5 Practice set a

2.7 Exercises for review

3 Addition of signed numbers

3.2 Addition of numbers with like signs

3.3 Sample set a

3.4 Practice set a

3.5 Addition with zero

3.6 Addition of numbers with unlike signs

3.7 Sample set b

3.8 Practice set b

3.10 Exercises for review

4 Subtraction of signed numbers

4.2 Definition of subtraction

4.3 Subtraction of signed numbers

4.4 Sample set a

4.5 Practice set a

4.7 Exercises for review

5 Multiplication and division of signed numbers

5.2 Multiplication of signed numbers

5.3 Sample set a

5.4 Practice set a

5.5 Division of signed numbers

5.6 Sample set b

5.7 Practice set b

5.8 Sample set c

5.9 Practice set c

5.11 Exercises for review

6 Negative exponents

6.2 Reciprocals

6.3 Negative exponents

6.4 Sample set a

6.5 Practice set a

6.6 Working with negative exponents

6.7 Sample set b

6.8 Practice set b

6.9 Sample set c

6.10 Practice set c

6.12 Exercises for review

7 Scientific notation

7.2 Standard form to scientific form

7.3 Sample set a

7.4 Practice set a

7.5 Scientific form to standard form

7.6 Sample set b

7.7 Practice set b

7.8 Working with numbers in scientific notation

7.9 Sample set c

7.10 Practice set c

7.12 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Signed numbers ( )

9.3 Absolute value ( )

9.4 Addition of signed numbers ( ) - multiplication and division of signed

9.5 Multiplication and division of signed numbers ( )

9.6 Negative exponents ( )

9.7 Scientific notation ( )

10 Proficiency exam

10.1 Proficiency exam

Algebraic expressions and equations

1 Algebraic expressions

1.2 Algebraic expressions

1.3 Terms and factors

1.4 Sample set a

1.5 Practice set a

1.6 Sample set b

1.7 Practice set b

1.8 Common factors

1.9 Sample set c

1.10 Practice set c

1.11 Coefficients

1.12 Sample set d

1.13 Practice set d

1.14 Sample set e

1.15 Practice set e

1.17 Exercises for review

2.3 Numerical evaluation

2.4 Sample set a

2.5 Practice set a

2.7 Exercises for review

3 Classification of expressions and equations

3.2 Polynomials

3.3 Classification of polynomials

3.4 Classification of polynomial equations

3.5 Sample set a

3.6 Practice set a

3.8 Exercises for review

4 Combining polynomials using addition and subtraction

4.3 Sample set a

4.4 Practice set a

4.5 Combining like terms

4.6 Sample set b

4.7 Practice set b

4.8 Simplifying expressions containing parentheses

4.9 Sample set c

4.10 Practice set c

4.12 Exercises for review

5 Combining polynomials using multiplication

5.2 Multiplying a polynomial by a monomial

5.3 Sample set a

5.4 Practice set a

5.5 Simplifying + ( a + b ) And - ( a + b )

5.6 Sample set b

5.7 Practice set b

5.8 Multiplying a polynomial by a polynomial

5.9 Sample set c

5.10 Practice set c

5.11 Sample set d

5.12 Practice set d

5.14 Exercises for review

6 Special binomial products

6.2 Expanding ( a + b ) 2 And ( a − b ) 2

6.3 Expanding ( a + b ) ( a − b )

6.4 Sample set a

6.5 Practice set a

6.7 Exercises for review

7 Terminology associated with equations

7.2 Independent and dependent variables

7.3 The domain of an equation

7.4 Sample set a

7.5 Practice set a

7.7 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Algebraic expressions ( )

9.3 Equations ( )

9.4 Classification of expressions and equations ( )

9.5 Combining polynomials using addition and subtraction ( ) - special

9.6 Terminology associated with equations ( )

10 Proficiency exam

10.1 Proficiency exam

Solving linear equations and inequalities

1 Solving equations

1.2 Types of equations

1.3 Solutions and equivalent equations

1.4 Sample set a

1.5 Practice set a

1.6 Literal equations

1.7 Solving equation of the form x + a = b and x − a = b

1.8 Sample set b

1.9 Practice set b

1.11 Exercises for review

2 Solving equations of the form ax=b and x/a=b

2.2 Equality property of division and multiplication

2.3 Solving a x = b And x a = b For x

2.4 Sample set a

2.5 Practice set a

2.7 Exercises for review

3 Further techniques in equation solving

3.2 Combining techniques in equation solving

3.3 Sample set a

3.4 Practice set a

3.5 Sample set b

3.6 Practice set b

3.7 Sample set c

3.8 Practice set c

3.9 Recognizing identities and contradictions

3.10 Sample set d

3.11 Practice set d

3.13 Exercises for review

4 Application i - translating from verbal to mathetical expressions

4.2 Translating from verbal to mathematical expressions

4.3 Sample set a

4.4 Practice set a

4.5 Sample set b

4.6 Practice set b

4.8 Exercises for review

5 Application ii - solving problems

5.2 Solving applied problems

5.3 Sample set a

5.4 Practice set a

5.5 Sample set b

5.6 Practice set b

5.7 Sample set c

5.8 Practice set c

5.9 Sample set d

5.10 Practice set d

5.11 Sample set e

5.12 Practice set e

5.14 Exercises for review

6 Linear inequalities in one variable

6.2 Inequalities

6.3 Linear inequalities

6.4 Sample set a

6.5 Practice set a

6.6 The algebra of linear inequalities

6.7 Sample set b

6.8 Practice set b

6.9 Compound inequalities

6.10 Sample set c

6.11 Practice set c

6.13 Exercises for review

7 Linear equations in two variables

7.2 Solutions to linear equations in two variables

7.3 Ordered pairs as solutions

7.4 Sample set a

7.5 Practice set a

7.7 Calculator problems

7.8 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Solving equations ( ) - further techniques in equation solving ( )

9.3 Application i - translating from verbal to mathetical expressions ( )

9.4 Application ii - solving problems ( )

9.5 Linear inequalities in one variable ( )

9.6 Linear equations in two variables ( )

10 Proficiency exam

10.1 Proficiency exam

Factoring polynomials

1 Finding the factors of a monomial

1.2 Products of polynomials

1.4 Sample set a

1.5 Practice set a

1.7 Exercises for review

2 Factoring a monomial from a polynomial

2.2 The factorization process

2.3 Sample set a

2.4 Practice set a

2.6 Exercises for review

3 The greatest common factor

3.2 Factoring method

3.3 Greatest common factor

3.4 Sample set a

3.5 Practice set a

3.6 Sample set b

3.7 Practice set b

3.9 Exercises for review

4 Factoring by grouping

4.2 Using grouping to factor a polynomial

4.3 Knowing when to try the grouping method

4.4 Sample set a

4.5 Practice set a

4.7 Exercises for review

5 Factoring two special products

5.2 The difference of two squares

5.3 Sample set a

5.4 Practice set a

5.5 Fundamental rules of factoring

5.6 Sample set b

5.7 Practice set b

5.8 Perfect square trinomials

5.9 Sample set c

5.10 Practice set c

5.12 Exercises for review

6 Factoring trinomials with leading coefficient 1

6.3 Sample set a

6.4 Practice set a

6.5 Factoring hints

6.6 Sample set b

6.7 Practice set b

6.9 Exercises for review

7 Factoring trinomials with leading coefficient other than 1

7.2 The method of factorization

7.3 The trial and error method of factoring a x 2 + b x + c

7.4 Sample set a

7.5 Practice set a

7.6 Sample set b

7.7 Practice set b

7.8 The collect and discard method of factoring a x 2 + b x + c

7.9 Sample set c

7.10 Practice set c

7.12 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Finding the factors of a monomial ( )

9.3 Factoring a monomial from a polynomial ( ) - the greatest common factor

9.4 Factoring by grouping ( )

9.5 Factoring two special products ( ) - factoring trinomials with leading

10 Proficiency exam

10.1 Proficiency exam

Graphing linear equations and inequalities in one and two variables

1 Graphing linear equations and inequalities in one variable

1.3 Axes, coordinate systems, and dimension

1.4 Graphing in one dimension

1.5 Sample set a

1.6 Practice set a

1.7 Sample set b

1.8 Practice set b

1.10 Exercises for review

2 Plotting points in the plane

2.3 Coordinates of a point

2.4 Plotting points

2.5 Sample set a

2.6 Practice set a

2.8 Exercises for review

3 Graphing linear equations in two variables

3.2 Solutions and lines

3.3 General form of a linear equation

3.4 The intercept method of graphing

3.5 Sample set a

3.6 Practice set a

3.7 Graphing using any two or more points

3.8 Sample set b

3.9 Practice set b

3.10 Slanted, horizontal, and vertical lines

3.11 Sample set c

3.12 Practice set c

3.14 Calculator problems

3.15 Exercises for review

4 The slope-intercept form of a line

4.2 The general form of a line

4.3 The slope-intercept form of a line

4.4 Sample set a

4.5 Practice set a

4.6 Slope and intercept

4.7 Sample set b

4.8 Practice set b

4.9 Sample set c

4.10 Practice set c

4.11 The formula for the slope of a line

4.12 Sample set d

4.13 Practice set d

4.15 Calculator problems

4.16 Exercises for review

5 Graphing equations in slope-intercept form

5.2 Using the slope and intercept to graph a line

5.3 Sample set a

5.4 Practice set a

5.6 Excersise for review

6 Finding the equation of a line

6.2 The slope-intercept and point-slope forms

6.3 Sample set a

6.4 Practice set a

6.5 Sample set b

6.6 Practice set b

6.7 Sample set c

6.8 Practice set c

6.10 Exercises for review

7 Graphing linear inequalities in two variables

7.2 Location of solutions

7.3 Method of graphing

7.4 Sample set a

7.5 Practice set a

7.7 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Graphing linear equations and inequalities in one variable ( )

9.3 Plotting points in the plane ( )

9.4 Graphing linear equations in two variables ( )

9.5 Graphing linear equations in two variables ( ) - graphing equations in

9.6 The slope-intercept form of a line ( )

9.7 Finding the equation of a line ( )

9.8 Graphing linear inequalities in two variables ( )

10 Proficiency exam

10.1 Proficiency exam

Rational expressions

1 Rational expressions

1.2 Rational expressions

1.3 Zero-factor property

1.4 Sample set a

1.5 Practice set a

1.6 The equality property of fractions

1.7 The negative property of fractions

1.8 Sample set b

1.9 Practice set b

1.11 Exercises for review

2 Reducing rational expressions

2.2 The logic behind the process

2.4 The process

2.5 Sample set a

2.6 Practice set a

2.8 Excercises for review

3 Multiplying and dividing rational expressions

3.2 Multiplication of rational expressions

3.3 Sample set a

3.4 Practice set a

3.5 Division of rational expressions

3.6 Sample set b

3.7 Practice set b

3.9 Exercises for review

4 Building rational expressions and the lcd

4.2 The process

4.3 Sample set a

4.4 Practice set a

4.5 The reason for building rational expressions

4.6 The least common denominator (lcd)

4.7 Sample set b

4.8 Practice set b

4.9 Sample set c

4.10 Practice set c

4.12 Excercises for review

5 Adding and subtracting rational expressions

5.3 Fractions with the same denominator

5.4 Sample set a

5.5 Practice set a

5.6 Fractions with different denominators

5.7 Sample set b

5.8 Practice set b

5.9 Sample set c

5.10 Practice set c

5.12 Exercises for review

6 Rational equations

6.2 Rational equations

6.3 Rational equations

6.4 The logic behind the process

6.5 The process

6.6 Extraneous solutions

6.7 Sample set a

6.8 Practice set a

6.9 Sample set b

6.10 Practice set b

6.11 Section 7.6 exercises

6.12 Exercises for review

7.2 The five-step method

7.3 Sample set a

7.4 Practice set a

7.5 Sample set b

7.6 Practice set b

7.7 Sample set c

7.8 Practice set c

7.9 Sample set d

7.10 Practice set d

7.11 Sample set e

7.12 Practice set e

7.13 Sample set f

7.14 Practice set f

7.16 Exercises for review

8 Complex rational expressions

8.2 Simple and complex fractions

8.3 The combine-divide method

8.4 Sample set a

8.5 Practice set a

8.6 The lcd-multiply-divide method

8.7 Sample set b

8.8 Practice set b

8.10 Exercises for review

9 Dividing polynomials

9.2 Dividing a polynomial by a monomial

9.3 Sample set a

9.4 Practice set a

9.5 The process of division

9.6 Review of subtraction of polynomials

9.7 Dividing a polynomial by a polynomial

9.8 Sample set b

9.9 Practice set b

9.10 Sample set c

9.11 Practice set c

9.13 Exercises for review

10 Summary of key concepts

10.1 Summary of key concepts

11 Exercise supplement

11.1 Exercise supplement

11.2 Rational expressions ( )

11.3 Reducing rational expressions ( )

11.4 Multiplying and dividing rational expressions ( ) - adding and subtracting

11.5 Rational equations ( )

11.6 Applications ( )

11.7 Dividing polynomials ( )

12 Proficiency exam

12.1 Proficiency exam

Roots, radicals, and square root equations

1 Square root expressions

1.2 Square roots

1.3 Sample set a

1.4 Practice set a

1.5 Principal and secondary square roots

1.6 Sample set b

1.7 Practice set b

1.8 Meaningful expressions

1.9 Sample set c

1.10 Practice set c

1.11 Simplifying square roots

1.12 Sample set d

1.13 Practice set d

1.15 Exercised for review

2 Simplifying square root expressions

2.2 Perfect squares

2.3 The product property of square roots

2.4 The quotient property of square roots

2.5 Square roots not involving fractions

2.6 Sample set a

2.7 Practice set a

2.8 Square roots involving fractions

2.9 Sample set b

2.10 Practice set b

2.12 Exercises for review

3 Multiplication of square root expressions

3.2 The product property of square roots

3.3 Multiplication rule for square root expressions

3.4 Sample set a

3.5 Practice set a

3.7 Exercises for review

4 Division of square root expressions

4.2 The division property of square roots

4.3 Rationalizing the denominator

4.4 Sample set a

4.5 Practice set a

4.6 Conjugates and rationalizing the denominator

4.7 Sample set b

4.8 Practice set b

4.10 Exercises for review

5 Addition and subtraction of square root expressions

5.2 The logic behind the process

5.3 The process

5.4 Sample set a

5.5 Practice set a

5.6 Sample set b

5.7 Practice set b

5.9 Exercises for review

6 Square root equations with applications

6.2 Square root equations and extraneous solutions

6.3 Method for solving square root equations

6.4 Sample set a

6.5 Practice set a

6.7 Exercises for review

7 Summary of key concepts

7.1 Summary of key concepts

8 Exercise supplement

8.1 Exercise supplement

8.2 Square root expressions ( ) - addition and subtraction of square root

8.3 Square root equations with applications ( )

9 Proficiency exam

9.1 Proficiency exam

Quadratic equations

1 Solving quadratic equations

1.2 Standard form of a quadratic equation

1.3 Sample set a

1.4 Practice set a

1.5 Zero-factor property

1.6 Sample set b

1.7 Practice set b

1.9 Exercises for review

2 Solving quadratic equations by factoring

2.2 Factoring method

2.3 Sample set a

2.4 Practice set a

2.5 Solving mentally after factoring

2.6 Sample set b

2.7 Practice set b

2.9 Exercises for review

3 Solving quadratic equations using the method of extraction of roots

3.2 The method of extraction of roots

3.3 The nature of solutions

3.4 Sample set a

3.5 Practice set a

3.6 Sample set b

3.7 Practice set b

3.9 calculator problems

3.10 Exercises for review

4 Solving quadratic equations using the method of completing the square

4.2 The logic behind the method

4.3 The method of completing the square

4.4 Sample set a

4.5 Practice set a

4.7 calculator problems

4.8 Exercises for review

5 Solving quadratic equations using the quadratic formula

5.2 Standard form of a quadratic equation

5.3 Sample set a

5.4 Practice set a

5.5 The quadratic formula

5.6 Derivation of the quadratic formula

5.7 Sample set b

5.8 Practice set b

5.10 Exercises for review

6.2 The five-step method

6.3 Sample set a

6.4 Practice set a

6.5 Sample set b

6.6 Practice set b

6.7 Sample set c

6.8 Practice set c

6.9 Sample set d

6.10 Practice set d

6.11 Sample set e

6.12 Practice set e

6.13 Sample set f

6.14 Practice set f

6.16 Sample set a—type problems

6.17 Sample set b—type problems

6.18 Sample set c—type problems

6.19 Sample set d—type problems

6.20 Sample set e—type problems

6.21 Sample set f—type problems

6.22 Astrophysical problem

6.23 Exercises for review

7 Graphing quadratic solutions

7.3 Constructing graphs of parabolas

7.4 Sample set a

7.5 Practice set a

7.6 Sample set b

7.7 Practice set b

7.9 Exercises for review

8 Summary of key concepts

8.1 Summary of key concepts

9 Exercise supplement

9.1 Exercise supplement

9.2 Solving quadratic equations ( ) - solving quadratic equations by

9.3 Solving quadratic equations using the method of extraction of roots (

9.4 Solving quadratic equations using the method of completing the square

9.5 Applications ( )

10 Proficiency exam

10.1 Proficiency exam

Systems of linear equations

1 Solutions by graphing

1.2 Systems of equations

1.3 Solution to a system of equations

1.4 Graphs of systems of equations

1.5 Independent, inconsistent, and dependent systems

1.6 The method of solving a system graphically

1.7 Sample set a

1.8 Practice set a

1.10 Exercises for review

2 Elimination by substitution

2.2 When substitution works best

2.3 The substitution method

2.4 Sample set a

2.5 Practice set a

2.6 Substitution and parallel lines

2.7 Sample set b

2.8 Practice set b

2.9 Substitution and coincident lines

2.10 Sample set c

2.11 Practice set c

2.12 Sample set d

2.13 Practice set d

2.15 Exercises for review

3 Elimination by addition

3.2 The properties used in the addition method

3.3 The addition method

3.4 Sample set a

3.5 Practice set a

3.6 Sample set b

3.7 Practice set b

3.8 Addition and parallel or coincident lines

3.9 Sample set c

3.10 Practice set c

3.12 Exercises for review

4.2 The five-step method

4.3 Sample set a (number problems)

4.4 Practice set a

4.5 Sample set b (coin problems)

4.6 Practice set b

4.7 Sample set c (mixture problems)

4.8 Practice set c

4.10 Exercises for review

5 Summary of key concepts

5.1 Summary of key concepts

6 Exercise supplement

6.1 Exercise supplement

6.2 Solutions by graphing ( ) - elimination by addition ( )

6.3 Applications ( )

7 Proficiency exam

7.1 Proficiency exam

Table of symbols

1 Properties of real numbers

1.1 Commutative property

1.2 Associative property

1.3 Distributive property

1.4 Properties of zero

1.5 Double negative property

2 Important and useful rules/formulas

2.1 Exponents (assume each expression is defined.)

2.2 Factorization and special product formulas

3 The 5-step method of solving applied problems

Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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