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Copyright © 2010 Pearson Education, Inc
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Algebraic Expressions
1 Introduction to Algebraic Expressions 1.1 Introduction to Algebra 1.2 The Commutative, Associative, and Distributive Laws 1.3 Fraction Notation 1.4 Positive and Negative Real Numbers 1.5 Addition of Real Numbers 1.6 Subtraction of Real Numbers 1.7 Multiplication and Division of Real Numbers 1.8 Exponential Notation and Order of Operations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Algebra
1.1 Introduction to Algebra Algebraic Expressions Translating to Algebraic Expressions Translating to Equations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Algebraic Expression An algebraic expression consists of
variables and/or numerals often with operation signs and grouping symbols. Examples: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Evaluate each expression for the given values.
Example Evaluate each expression for the given values. a) a + b for a = 21 and b = 74 Solution a + b = = 95 b) 7xy for x = 3 and y = 6 Solution 7xy = 7 • 3 • 6 = 126 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example The area of a triangle with a base length b and height of length h is given by the formula A = ½ bh. Find the area when b is 12 m and h is 7.2 m. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Translating to Algebraic Expressions
per of decreased by increased by ratio of twice less than more than divided into times minus plus quotient of product of difference of sum of divided by multiplied by subtracted from added to Division Multiplication Subtraction Addition Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Translate each phrase to an algebraic expression.
a) 9 more than y b) 7 less than x c) the product of 3 and twice w Solution Phrase Algebraic Expression a) 9 more than y y + 9 b) 7 less than x x  7 c) the product of 3 and twice w 3•2w or 2w • 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Translating to Equations
The symbol = (“equals”) indicates that the expressions on either side of the equals sign represent the same number. An equation is a number sentence with the verb =. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution A replacement or substitution that makes an equation true is called a solution. Some equations have more than one solution, and some have no solution. When all solutions have been found, we have solved the equation. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Determine whether 12 is a solution of x + 4 = 16. Solution
x + 4 = 16 Writing the equation | Substituting 12 for x 16 = = 16 is TRUE. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Translate the problem to an equation.
What number added to 127 is 403? Solution Let y represent the unknown number. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Commutative, Associative, and Distributive Laws
1.2 The Commutative, Associative, and Distributive Laws Equivalent Expressions The Commutative Laws The Associative Laws The Distributive Law The Distributive Law and Factoring Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Commutative Laws a + b = b + a.
For Addition. For any numbers a and b, a + b = b + a. (Changing the order of addition does not affect the answer.) For Multiplication. For any numbers a and b, ab = ba. (Changing the order of multiplication does not affect the answer.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the commutative laws to write an expression equivalent to each of the following. a) r + 7 b) 12y c) 9 + st Solution a) r + 7 is equivalent to 7 + r b) 12y is equivalent to y • 12 c) 9 + st is equivalent to st + 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Associative Laws For Addition. For any numbers a, b, and c,
a + (b + c) = (a + b) + c. (Numbers can be grouped in any manner for addition.) For Multiplication. For any numbers a, b, and c, a • (b • c) = (a • b) • c. (Numbers can be grouped in any manner for multiplication.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

a) t + (4 + y) is equivalent to (t + 4) + y
Example Use the associative laws to write an expression equivalent to each of the following. a) t + (4 + y) b) (12y)z Solution a) t + (4 + y) is equivalent to (t + 4) + y b) (12y)z is equivalent to 12(yz) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Distributive Law For any numbers a, b, and c, a(b + c) = ab + ac.
(The product of a number and a sum can be written as the sum of two products.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiply: 4(x + 7) Solution Example
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Multiply: 7(x + y + 4z) Solution Example
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Multiply: (a + 3)2 Solution Example
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The Distributive Law and Factoring
If we use the distributive law in reverse, we have the basis of a process called factoring. To factor an expression means to write an equivalent expression that is a product. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the distributive law to factor each of the following. a) 5x + 5y b) 8y + 32w + 8 Solution a) 5x + 5y 5x + 5y = 5(x + y) b) 8y + 32w + 8 = 8y + 84w + 81 = 8(y + 4w + 1) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1.3 Fraction Notation Factors and Prime Factorizations
Multiplication, Division, and Simplification More Simplifying Addition and Subtraction Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factors and Prime Factorizations
Natural Numbers can be thought of as the counting numbers: 1, 2, 3, 4, 5… (The dots indicated that the established pattern continues without ending.) To factor a number, we simply express it as a product of two or more numbers. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Write several factorizations of 18. Then list all the factors of 18. Solution The factors of 18 are: 1, 2, 3, 6, 9, and 18. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Prime Number A prime number is a natural number that has exactly two different factors: the number itself and 1. The first several primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Definitions If a natural number, other than 1, is not prime, we call it composite. Every composite number can be factored into a product of prime numbers. Such a factorization is called the prime factorization of that composite number. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the prime factorization of 48. Solution
The prime factorization of 48 is Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Fraction Notation The top number is called the numerator and the bottom number is called the denominator. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

(Any nonzero number divided by itself is 1.)
Fraction Notation for 1 For any number a, except 0, (Any nonzero number divided by itself is 1.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiplication of Fractions
For any two fractions a/b and c/d, (The numerator of the product is the product of the two numerators . The denominator of the product is the product of the two denominators.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply: a) b) Solution a) b)
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Two numbers whose product is 1 are reciprocals, or multiplicative inverses.
The reciprocal of is because Example Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Division of Fractions To divide two fractions, multiply by the reciprocal of the divisor: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Divide: Example Solution
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The Identity Property of 1
For any number a, a ● 1 = 1 ● a = a. (Multiplying a number by 1 gives the same number.) The number 1 is called the multiplicative identity. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution
Factoring the numerator and the denominator using a common factor of 5. Rewriting as a product of two fractions Using the identity property of 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Addition and Subtraction of Fractions
For any two fractions a/d and b/d, Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Add and simplify: Solution Using 60 as the common denominator
Perform the multiplication Adding fractions & simplifying Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Perform the indicated operation and, if possible, simplify.
Solution Removing a factor equal to 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Perform the indicated operation and, if possible, simplify.
Solution Removing a factor equal to 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Positive and Negative Real Numbers
1.4 Positive and Negative Real Numbers The Integers The Rational Numbers Real Numbers and Order Absolute Value Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

A set is a collection of objects.
The integers consist of all whole numbers and their opposites. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The set of integers = {…−4, −3, −2, −1, 0, 1, 2, 3, 4, …}
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Example State which integer(s) corresponds to the situation.
The lowest point in Australia is Lake Eyre at 15 m below sea level and the highest point is Mt. Kosciuszko at 2229 m above sea level. Solution The integer −15 corresponds to 15 m below sea level. The integer 2229 corresponds to 2229 m above sea level. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Set of Rational Numbers The set of rational numbers =
This is read “the set of all numbers a over b, where a and b are integers and b does not equal zero.” Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Convert to decimal notation: Solution ← The remainder is 0.
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Example Convert to decimal notation: Solution
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Set of Real Numbers The set of real numbers = The set of all numbers corresponding to points on the number line. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Example Solution a) whole numbers: 0, 4, 65, 72
Which numbers in the following list are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers? Solution a) whole numbers: 0, 4, 65, 72 b) integers: −27, 0, 4, 65, 72 c) rational numbers: d) irrational numbers: e) real numbers: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

> means “greater than”
Real numbers are named in order on the number line, with larger numbers further to the right. < mean “less than” > means “greater than” −5 < 8 10 -9 -7 -5 -3 -1 1 3 5 7 9 -10 -8 -4 4 8 -2 6 -6 2 −2 > −9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use either < or > for the blank to write a true statement. (a) −3.42 ____ 2.35 (b) 7 ____ −15 (c) −11 ____ −9 Solution a) Since −3.42 is to the left of 2.35, we have −3.42 < 2.35. b) Since 7 is to the right of −15, we have 7 > −15. c) Since −11 is to the left of −9, we have −11 < −9. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More Inequalities ≤ means “is less than or equal to”
 means “is greater than or equal to” Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Classify each inequality as true or false.
a) −9 ≤ 7 b) −8  −8 c) −7  −2 Solution a) −9 ≤ 7 is true because −9 < 7. b) −8  −8 is true because −8 = −8. c) −7  −2 is false since −7 > −2 nor −7 = −2 is true. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Absolute Value We write |a|, read “the absolute value of a,” to represent the number of units that a is from zero. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Find the absolute value: |–4| |3.8| |0|
a) |–4| = 4 since –4 is 4 units from 0. b) |3.8| = 3.8 since 3.8 is 3.8 units from 0. c) |0| = 0 since 0 is 0 units from itself. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Addition of Real Numbers
1.5 Addition of Real Numbers Adding with the Number Line Adding Without the Number Line Problem Solving Combining Like Terms Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Adding with a Number Line
To add a + b on a number line, we start at a and move according to b. a) If b is positive, we move to the right (the positive direction). b) If b is negative, we move to the left (the negative direction). c) If b is 0, we stay at a. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Add: −3 + 7. Solution
Locate the first number −3, and then move 7 units to the right 10 -9 -7 -5 -3 -1 1 3 5 7 9 -10 -8 -4 4 8 -2 6 -6 2 Start at −3 Move 7 units to the right. −3 + 7 = 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Add: −2 + (−3). Solution
After locating −2, we move 3 units to the left. 10 -9 -7 -5 -3 -1 1 3 5 7 9 -10 -8 -4 4 8 -2 6 -6 2 Start at −2 Move 3 units to the left. −2 + (−3) = −5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rules for Addition of Real Numbers
1. Positive numbers: Add as usual. The answer is positive. 2. Negative numbers: Add the absolute values and make the answer negative. 3. A positive and a negative number: Subtract the smaller absolute value from the greater absolute value. a) If the positive number has the greater absolute value, the answer is positive. b) If the negative number has the greater absolute value, the answer is negative. c) If the numbers have the same absolute value, the answer is 0. 4. One number is zero: The sum is the other number. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Adding Without the Number Line
a) −9 + (−11) Two negatives, add the absolute value, answer is −20. b) − A negative and a positive, subtract and the answer is negative, −19. c) − A negative and a positive, subtract and the answer is positive, 5.1. d) (−2.4) A negative and a positive, subtract and the answer is 0. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Add: 17 + (−3) + 9 + 16 + (−4) + (−12).
= (−3)+ (−4) + (−12) Using the commutative law = ( ) +[(−3)+ (−4) + (−12)] Using the associative law = 42 + (−19) = 23 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Problem Solving During the first two weeks of the semester, 6 students withdrew from Mr. Lange’s algebra class, 9 students were added to the class, and 4 students were dropped as “no-shows.” By how many students did the original class size change? The original class size dropped by one. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Combining Like Terms Example Solution −5x + 7x = (−5 + 7)x
When two terms have variable factors that are exactly the same, the terms are called like or similar terms. Combine like terms −5x + 7x. Solution −5x + 7x = (−5 + 7)x = 2x Example Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Combine like terms: 3a + (−4b) + (−8a) + 10b
= 3a + (−8a) + (−4b) + 10b = (3 +(−8))a + (−4 + 10)b = −5a + 6b Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Subtraction of Real Numbers
1.6 Subtraction of Real Numbers Opposites and Additive Inverses Subtraction Problem Solving Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Opposite The opposite, or additive inverse, of a number a is written −a (read “the opposite of a” or “the additive inverse of a”). Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Opposite of an Opposite
For any real number a, −(−a) = a. (The opposite of the opposite of a is a.) Example Find −x and −(−x) when x = 27. Solution If x = 27, then −x = −27. If x = 27, then −(−27) = 27. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Law of Opposites For any two numbers a and –a, a + (–a) = 0.
(When opposites are added, their sum is 0.) The opposite of 4 is –4. The opposite of 0 is 0. The opposite of –15 is 15. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Subtraction of Real Numbers
For any real numbers a and b, a – b = a + (–b) (To subtract, add the opposite, or additive inverse, of the number being subtracted.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Subtract each of the following.
a) 3  b) 8 – (–6) c) –3.4 – (–2.6) d) Solution a) 3 – 7 = 3 + (−7) = −4 b) 8 – (–6) = = 14 c) –3.4 – (–2.6) = – = –0.8 d) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: 9 − (−3) − 5 − (−8) + 4. Solution
9 − (−3) − 5 − (−8) + 4 = (−5) = 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Combine like terms. a) 4 + 5x − 19x b) −7a − 4b − 3a + 12b
Solution a) 4 + 5x − 19x = 4 + 5x + (−19x) = 4 + (5 + (− 19))x = 4 + (− 14)x = 4 − 14x b) −7a − 4b − 3a + 12b = −7a + (−4b) + (−3a) + 12b = −7a + (−3a) + (−4b) + 12b = −10a + 8b Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Problem Solving Example Solution
The lowest point in Australia is Lake Eyre at 15 m below sea level and the highest point is Mt. Kosciuszko at 2229 m above sea level. What is the difference in elevation? Solution Higher elevation − Lower elevation = 2229 − (−15) = 2244 m The difference in elevation is 2244 m. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiplication and Division of Real Numbers
1.7 Multiplication and Division of Real Numbers Multiplication Division Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Product of a Negative Number and a Positive Number
To multiply a positive number and a negative number, multiply their absolute values. The answer is negative. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply: 9(−3) = −27 b)−4(12) = −48 c)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Multiplicative Property of Zero
For any real number a, 0  a = a  0 = 0. (The product of 0 and any real number is 0.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply: 125(−721)(0)
= 125 [(−721)(0)] Using the associative law = 125(0) Using the multiplicative property of 0 = 0 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Product of Two Negative Numbers
To multiply two negative numbers, multiply their absolute values. The answer is positive. Example a) −9(−9) = 9 · 9 = 81 b) −4(−12) = 4 · 12 = 48 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

When three or more numbers are multiplied, we can order the numbers as we please, because of the commutative and associative laws. Multiply a) −4(−5)(−6) b) −5(−6)(−2)(−3) Solution a) −4(−5)(−6) = 20(−6) = −120 b) −5(−6)(−2)(−3) = 30  6 = 180 The product of an even number of negative numbers is positive. The product of an odd number of negative numbers is negative. Example Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Divide, if possible. a) 21 ÷ (−3) b) c)
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Rules for Multiplication and Division
To multiply or divide two nonzero real numbers: 1. Using the absolute values, multiply or divide, as indicated. 2. If the signs are the same, the answer is positive. 3. If the signs are different, the answer is negative. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example a. Divide: Solution b. Divide: Solution
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Division Involving Zero
For any real number a, and for a ≠ 0, Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Exponential Notation and Order of Operations
1.8 Exponential Notation and Order of Operations Exponential Notation Order of Operations Simplifying and the Distributive Law The Opposite of a Sum Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The 5 is called an exponent. The 4 is the base.
Exponential Notation The 5 is called an exponent. The 4 is the base. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Write exponential notation for 777777.
Solution Exponential notation is 76 7 is the base. 6 is the exponent. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Simplify: a) 82 b) (−8)3 c) (4y)3 a) 82 = 8  8 = 64
= 64(−8) = − 512 c) (4y)3 = (4y) (4y) (4y) = 4  4 4 y y y = 64y3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Exponential Notation For any natural number n,
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Rules for Order of Operations
1. Calculate within the innermost grouping symbols, ( ), [ ], { }, | |, and above or below fraction bars. 2. Simplify all exponential expressions. 3. Perform all multiplication and division working from left to right. 4. Perform all addition and subtraction working from left to right. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution Multiplying
Subtracting and adding from left to right Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Evaluate 16  8(7  y)2 for y = 2. Solution
= 16  8(5)2 = 16  8(25) = 2(25) = 50 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution
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Example Simplify: Solution
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Simplifying and the Distributive Law
Example Simplify: Solution Distributive Law Combining Like Terms Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Property of −1 For any real number a, −1  a = −a.
(Negative one times a is the opposite of a.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Write an expression equivalent to (4x + 3y + 5) without using parentheses. Solution (4x + 3y + 5) = 1(4x + 3y + 5) = 1(4x) + 1(3y) + 1(5) = 4x  3y  5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Opposite of a Sum For any real numbers a and b,
−(a + b) = −a + (−b) = −a − b (The opposite of a sum is the sum of the opposites.) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Simplify: Solution
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Example Simplify: Solution
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