cei03 ppt 01


Foundations of AlgebraCHAPTER

1.1 Number Sets and the Structure of Algebra1.2 Fractions1.3 Adding and Subtracting Real Numbers;

Properties of Real Numbers1.4 Multiplying and Dividing Real Numbers;

Properties of Real Numbers1.5 Exponents, Roots, and Order of Operations1.6 Translating Word Phrases to Expressions1.7 Evaluating and Rewriting Expressions

11


Number Sets and the Structure of Algebra1.11.1

1. Understand the structure of algebra.2. Classify number sets.3. Graph rational numbers on a number line.4. Determine the absolute value of a number.5. Compare numbers.

4Copyright © 2011 Pearson Education, Inc.

Objective 1

Understand the structure of algebra.


Definitions

Variable: A symbol that can vary in value.Constant: A symbol that does not vary in value.

Variables are usually letters of the alphabet, like x or y.Usually constants are symbols for numbers, like 1, 2, ¾, 6.74.


Expression: A constant, variable, or any combination of constants, variables, and arithmetic operations that describe a calculation.

Examples of expressions:

2 + 6 or 4x 5 or 21

3 r h


Equation: A mathematical relationship that contains an equal sign.

Examples of equations:

2 + 6 = 8 or 4x 5 = 12 or 21

3V r h


Inequality: A mathematical relationship that contains an inequality symbol (, <, >, , or ).

Symbolic form Translation

8 3 Eight is not equal to three.

5 < 7 Five is less than seven.

7 > 5 Seven is greater than five.

x 3 x is less than or equal to three.

y 2 y is greater than or equal to two.


Objective 2

Classify number sets.


Set: A collection of objects.

Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 would be written {1, 2, 3, 4}.

The numbers 1, 2, 3, and 4 are called the members or elements of this set.


Writing SetsTo write a set, write the members or elements of the set separated by commas within braces, { }.


Example 1

Write the set containing the first four days of the week.

Answer{Sunday, Monday, Tuesday, Wednesday}


Numbers are classified using number sets.

Natural numbers contain the counting numbers 1, 2, 3, 4, …and is written {1, 2, 3, …}. The three dots are called ellipsis and indicate that the numbers continue forever in the same pattern.

Whole numbers: natural numbers and 0 {0, 1, 2, 3,…}Integers: whole numbers and the opposite (or negative) of every natural number {…, 3, 2, 1, 0, 1, 2, 3…}

Rational: every real number that can be expressed as a ratio of integers.


Rational number: Any real number that can be expressed in the form , where a and b are integers and b 0.

a

b


Example 2

Determine whether the given number is a rational number.a. b. 0.8 c.

Answera.

5

60.3

5

6

Yes, because 5 and 6 are integers.

b. 0.8

Yes, 0.8 can be expressed as a fraction 8 over 10, and 8 and 10 are integers.

c. 0.3

The bar indicates that the digit repeats. This is the decimal equivalent of 1 over 3. Yes this is a rational number.


Irrational number: Any real number that is not rational.

Examples:

Real numbers: The union of the rational and irrational numbers.

2, 3,


Objective 3

Graph rational numbers on a number line.


Example 3

Graph on a number line.

Answer The number is located 4/5 of the way between 2 and 3.

42

5

31-1 20 42

5

Between 2 and 3, we divide the number line into 5 equally spaced divisions. Place a dot on the 4th mark.


Objective 4

Determine the absolute value of a number.


Absolute value: A given number’s distance from 0 on a number line.

The absolute value of a number n is written |n|.The absolute value The absolute valueof 5 is 5. of 5 is 5.|5| = 5 |5| = 5

5 units from 0 5 units from 0


Absolute ValueThe absolute value of every real number is either positive or 0.


Example 4

Simplify.a. |9.4| b.

Answer

a. |9.4| = 9.4

b.

2

9

2

9

2

9


Objective 5

Compare numbers.


Comparing NumbersFor any two real numbers a and b, a is greater than b if a is to the right of b on a number line. Equivalently, b is less than a if b is to the left of a on a number line.


Example 5

Use =, <, or > to write a true statement.a. 3 ___ 3 b. 1.8 ___ 1.6

Answera. 3 ___ 3

3 > 3because 3 is farther right on a number line.

b. 1.8 ___ 1.6 1.8 < 1.6because –1.8 is further to the left on a number line.


To which set of numbers does 6 belong?

a) Irrational

b) Natural and whole numbers

c) Natural numbers, whole numbers,

and integersd) Integers and rational numbers

1.1


Simplify |7|.

a) 7

b) 7

c) 0

d) 1/7

1.1


Which statement is false?

a) 7 > 4

b) 2.4 > 1.4

c) 10 < 22

d) 3.6 > 6.4

1.1


Fractions1.21.2

1. Write equivalent fractions.2. Write equivalent fractions with the LCD.3. Write the prime factorization of a number.4. Simplify a fraction to lowest terms.


Fraction: A quotient of two numbers or expressions a and b having the form where b 0.

The top number in a fraction is called the numerator.The bottom number is called the denominator.Fractions indicated part of a whole.

,a

b

3

4

Numerator

Denominator


Objective 1

Write equivalent fractions.


Writing Equivalent FractionsFor any fraction, we can write an equivalent fraction by multiplying or dividing both its numerator and denominator by the same nonzero number.


Example 1

Find the missing number that makes the fractions equivalent.a. b.

Solutiona. b.

9

15 5

?

4

?18

36 2

9 ?

15 45

9 3

315

27

45

18 ?

36 2

18

36

18

18

1

2

Multiply the numerator and denominator by 3.

Divide the numerator and denominator by 6.


Objective 2

Write equivalent fractions with the LCD.


Multiple: A multiple of a given integer n is the product of n and an integer.

We can generate multiples of a given number by multiplying the given number by the integers.

2 1 2

2 2 4

2 3 6

2 4 8

2 5 10

2 6 12

3 1 3

3 2 6

3 3 9

3 4 12

3 5 15

3 6 18

Multiples of 2 Multiples of 3


Least common multiple (LCM): The smallest number that is a multiple of each number in a given set of numbers.

Least common denominator (LCD): The least common multiple of the denominators of a given set of fractions.


Example 2

Write as equivalent fractions with the LCD.

SolutionThe LCD of 8 and 6 is 24.

7 5 and

8 6

3

7 7=

8

3

8

21

24

4

5 5=

6

4

6

20

24


Objective 3

Write the prime factorization of a number.


Factors: If a b = c, then a and b are factors of c.

Example: 6 7 = 42, 6 and 7 are factors of 42

Prime number: A natural number that has exactly two different factors, 1 and the number itself.

Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…

Prime factorization: A factorization that contains only prime factors.


Example 3

Find the prime factorization of 420.Solution 420

42

10

6

7

2

5

2

3

Factor 420 to 10 and 42. (Any two factors will work.)

Factor 10 to 2 and 5, which are primes. Then factor 42 to 6 and 7.

7 is prime and then factor 6 into 2 and 3, which are primes.

Answer 2 2 3 5 7


Objective 4

Simplify a fraction to lowest terms.


Lowest terms: Given a fraction and b 0, if the only factor common to both a and b is 1, then the fraction is in lowest terms.

a

b


Simplifying a Fraction with the Same Nonzero Numerator and Denominator

Eliminating a Common Factor in a Fraction

11, when 0.

1

nn

n

1, when 0 and 0.

1

an a ab n

bn b b


These rules allow us to write fractions in lowest terms using prime factorizations. The idea is to replace the numerator and denominator with their prime factorizations and then eliminate the prime factors that are common to both the numerator and denominator.


Simplifying a Fraction to Lowest Terms

To simplify a fraction to lowest terms:1. Replace the numerator and denominator with their

prime factorizations.2. Eliminate (divide out) all prime factors common to the numerator and denominator.3. Multiply the remaining factors.


Example 4a

Simplify to lowest terms.

Solution

30

42

30

42

2 3 5

2 3 7

Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.

5

7


Example 4b

Simplify to lowest terms.

Solution

220

2380

220

2380

2 2 5 11

2 2 5 7 17

Replace the numerator and denominator with their prime factorizations; then eliminate the common prime factors.

11

119


Example 5

At a company, 225 of the 1050 employees have optional eye insurance coverage as part of their benefits package. What fraction of the employees have optional eye insurance coverage?Solution

Answer 3 out of 14 employees have optional eye insurance.

225

1050

3 3 5 5

2 3 5 5 7

3

14


What is the prime factorization of 360?

a) 6 6 5

b) 23 32 5

c) 22 32 5

d) 32 5 7

1.2


Simplify to lowest terms:

a)

b)

c)

d)

112

280

14

35

2

5

1

4

21

23

1.2


Adding and Subtracting Real Numbers; Properties of Real Numbers1.31.3

1. Add integers.2. Add rational numbers.3. Find the additive inverse of a number.4. Subtract rational numbers.


Objective 1

Add integers.


Parts of an addition statement: The numbers added are called addends and the answer is called a sum.

2 + 3 = 5

Addends Sum


Properties of Addition

Symbolic Form Word Form

Additive Identity

a + 0 = a The sum of a number and 0 is that number.

Commutative Property of Addition

a + b = b + a Changing the order of addends does not affect the sum.

Associative Property of Addition

a + (b + c) = (a + b) + c Changing the grouping of three or more addends does not affect the sum.


Example 1Indicate whether each equation illustrates the additive identity, commutative property of addition, or the associative property of addition.a. (5 + 6) + 3 = 5 + (6 + 3)Answer Associative property of addition

b. 0 + (9) = 9Answer Additive identity

c. (9 + 6) + 4 = 4 + (9 + 6) Answer Commutative property of addition


Adding Numbers with the Same Sign

To add two numbers that have the same sign, add their absolute values and keep the same sign.


Example 2

Add.a. 27 + 12 b. –16 + (– 22)

Solutiona. 27 + 12 = 39

b. –16 + (–22) = –38


Adding Numbers with Different Signs

To add two numbers that have different signs, subtract the smaller absolute value from the greater absolute value and keep the sign of the number with the greater absolute value.


Example 3

Add.a. 35 + (–17) b. –29 + 7

Solutiona. 35 + (–17) = 18

b. –29 + 7 = –22


Example 3 continued

Add.c. 15 + (–27) d. –32 + 6

Solutionc. 15 + (–27) = –12

d. –32 + 6 = –26


Objective 2

Add rational numbers.


Adding Fractions with the Same Denominator

To add fractions with the same denominator, add the numerators and keep the same denominator; then simplify.


Example 4

Add.a. b.

Solutiona.

2 4

9 9

2 4

9 9

2 3 2

3 3 3

4 5

12 12

4 5b.

12 12

3 3 3

3 2 2 4

Replace 6 and 9 with their prime

factorizations, divide out the common factor, 3, then multiply the remaining factors.

Simplify to lowest terms by dividing out the common factor, 3.

2 4 6

9 9

4 5 9

12 12


Example 4 continued

Add.c.

Solutiona.

7 3

10 10

7 3

10 10

2 2 2

2 5 5

Simplify to lowest terms by dividing out the common factor, 2.

7 ( 3) 4

10 10


Adding Fractions

To add fractions with different denominators:1. Write each fraction as an equivalent fraction with

the LCD.2. Add the numerators and keep the LCD.3. Simplify.

Solution

Slide 1- 71Copyright © 2011 Pearson Education, Inc.

1 4 1(3)

3 4 4(3)

Example 5a1 1

Add: 3 4

1 1

3 4 Write equivalent fractions

with 12 in the denominator.

4 3

12 12 Add numerators and keep

the common denominator.

7

12

Because the addends have the same sign, we add and keep the same sign.

Solution


5 2 3(3)

6 2 4(3)

Example 5b5 3

Add: 6 4

5 3



10 9



10 9

12

Because the addends have different signs, we subtract and keep the sign of the number with the greater absolute value.

1

12

Solution


7 15 9(4)

8 15 30(4)

Example 5c7 9

Add: 8 30

7 9



105 36



105 36

120

Reduce to lowest terms.

69

120

3 23

2 2 2 3 5

23

40

Anna has a balance of $378.45 and incurs a debt of $85.42. What is Anna’s new balance?

Solution

A debt of $85.42 is $85.42. Her balance is 378.45 + (– 85.42) = $293.03


Example 6


Objective 3

Find the additive inverse of a number.


Additive inverses: Two numbers whose sum is 0.

What happens if we add two numbers that have the same absolute value but different signs, such as 5 + (–5)? In money terms, this is like making a $5 payment towards a debt of $5. Notice the payment pays off the debt so that the balance is 0.

5 + (–5) = 0

Because their sum is zero, we say 5 and –5 are additive inverses, or opposites.


Example 7

Find the additive inverse of the given number.a. 8 b. –2 c. 0

Answersa. –8 because 8 + (–8) = 0

b. 2 because – 2 + 2 = 0

c. 0 because 0 + 0 = 0


Example 8

Simplify.a. – (–5) b. –|2| c. –| –9|

Answersa. – (–5) = 5

b. –|2| = –2

c. –| –9| = –9


Objective 4

Subtract rational numbers.


Parts of a subtraction statement:

8 – 5 = 3

Minuend

Subtrahend

Difference


Rewriting Subtraction

To write a subtraction statement as an equivalent addition statement, change the operation symbol from a minus sign to a plus sign, and change the subtrahend to its additive inverse.


Example 9a

Subtracta. –17 – (–5)

SolutionWrite the subtraction as an equivalent addition.

–17 – (–5)

= –17 + 5 = –12

Change the operation from minus to plus.

Change the subtrahend to its additive inverse.

Solution


Example 9b3 1

Subtract: 8 4

3 1

8 4

1

4

3

8

Write equivalent fractions with the common denominator, 8.

3 1(2)

8 4(2)

3 2

8 8

1

4

3

8

5

8


Example 9c

c. 4.07 – 9.03

Solution Write the equivalent addition statement.

4.07 – 9.03 = 4.07 + (– 9.03) = –4.96


Example 10

In an experiment, a mixture begins at a temperature of 52.6C. The mixture is then cooled to a temperature of 29.4C. Find the difference between the initial and final temperatures.

Solution 52.6 – (–29.4) = 52.6 + 29.4 = 82

Answer The difference between the initial and final temperatures is 82C.


Add –6 + (–9).

a) –15

b) 3

c) 3

d) 15

1.3


Subtract 5 – (–8).

a) –13

b) 3

c) 3

d) 13

1.3


Subtract

3 1.

7 3

a)

b)

c)

d)

16

21

1

2

2

21

2

21

1.3


Subtract

3 1.

7 3

a) 16

21

b) 1

2

c) 2

21

d) 2

21

1.3


Multiplying and Dividing Real Numbers; Properties of Real Numbers1.41.4

1. Multiply integers.2. Multiply more than two numbers.3. Multiply rational numbers.4. Find the multiplicative inverse of a number.5. Divide rational numbers.


Objective 1

Multiply integers.


In a multiplication statement, factors are multiplied to equal a product.

ProductFactors

2 3 = 6


Properties of Multiplication

Symbolic Form Word Form

Multiplicative Property of 0

The product of a number multiplied by 0 is 0.

Multiplicative Identity

The product of a number multiplied by 1 is the number.

Commutative Property of

Multiplication

ab=ba Changing the order of factors does not affect the product.

Associative Property of

Multiplication

a(bc) = (ab)c Changing the grouping of three or more factors does not affect the product.

Distributive Property of

Multiplication over Addition

a(b + c) =ab + ac A sum multiplied by a factor is equal to the sum of that factor multiplied by each addend.

0 0 a

1 a a


Example 1Give the name of the property of multiplication that is illustrated by each equation.a. 6(3) = 3 6Answer Commutative property of multiplication

b. 3(9 5) = [3(9)] 5Answer Associative property of multiplication

c. 4(4 – 2) = 4 4 – 4 2 Answer Distributive property of multiplication over addition


Multiplying Two Numbers with Different Signs

When multiplying two numbers that have different signs, the product is negative.


Example 2

Multiply.

a. 7(–4) b. (–15)3

Solutiona. 7(–4) =

b. (–15)3 =

Warning: Make sure you see the difference between 7(–4), which indicates multiplication, and 7 – 4, which indicates subtraction.

–28

–45


Multiplying Two Numbers with the Same Sign

When multiplying two numbers that have the same sign, the product is positive.


Example 3

Multiply.

a. –5(–9) b. (–6)(–8)

Solutiona. –5(–9) =

b. (–6)(–8) =

45

48


Objective 2

Multiply more than two numbers.


Multiplying with Negative Factors

The product of an even number of negative factors is positive, whereas the product of an odd number of negative factors is negative.


Example 4

Multiply.a. (–1)(–3)(–6)(7) Solution Because there are three negative factors (an

odd number of negative factors), the result is negative. (–1)(–3)(–6)(7) = –126

b. (–2)(–4)(2)(–5)(–3)Solution Because there are four negative factors(an

even number of negative factors), the result is positive. (–2)(–4)(2)(–5)(–3) = 240


Objective 3

Multiply rational numbers.


Multiplying Fractions

, where 0 and 0.a c ac

b db d bd


Example 5a

Multiply

Solution

3 4 .

5 9

3 4 3 2 2

5 9 5 3 3

4

15

Divide out the common factor, 3.

Because we are multiplying two numbers that have different signs, the product is negative.


Example 5b

Multiply

Solution

6 6 12

15 16 15

3

25

Divide out the common factors.

Because there are an even number of negative factors, the product is positive.

6 6 12

15 16 15

2 3 2 3 2 2 3

3 5 2 2 2 2 3 5


Multiplying Decimal Numbers

To multiply decimal numbers:1. Multiply as if they were whole numbers.2. Place the decimal in the product so that it has the

same number of decimal places as the total number of decimal places in the factors.


Example 6a

Multiply (–7.6)(0.04).Solution First, calculate the value and disregard signs for now.

0.04 2 places 7.6 + 1 place 0 2 4+ 0 2 8 0 0.3 0 4

Answer –0.304

When we multiply two numbers with different signs, the product is negative.

3 places


Example 6b

Multiply (3)(5.2)(1.4)(6.1).Solution First, calculate the value and disregard signs for now.

Multiply from left to right.

(3)(5.2)(1.4)(6.1) = (15.6)(1.4)(6.1) = 21.84(6.1) = 133.224

Answer 133.224

15.6 = (3)(5.2)

21.84 = 15.6(1.4)

The product of an even number of negative factors is positive. The factors have a total of 3 decimal places, so the product has three decimal places.


Objective 4

Find the multiplicative inverse of a number.


Multiplicative inverses: Two numbers whose product is 1.

2

3 and are multiplicative inverses because their product is 1.

3

2

2 3 6 1

3 2 6

Notice that to write a number’s multiplicative inverse, we simply invert the numerator and denominator. Multiplicative inverses are also known as reciprocals.


Example 7

Find the multiplicative inverse.a. b. c. 9

Answera. The multiplicative inverse is

b. The multiplicative inverse is 8.

c. The multiplicative inverse is

2

7

7.

2

1.

9

1

8


Objective 5

Divide rational numbers.


Dividend

8 2 = 4

Divisor

Quotient

Parts of a division statement:


Dividing Signed Numbers

When dividing two numbers that have the same sign, the quotient is positive.When dividing two numbers that have different signs, the quotient is negative.


Example 8

Divide.a. b.

Solutiona. b.

56 ( 8)

56 ( 8) 7

72 6

72 6 12


Division Involving 0

0 0 when 0.n n

0 is undefined when 0.n n

0 0 is indeterminate.


Dividing Fractions

, where 0, 0, and 0.a c a d

b c db d b c


Example 9

Divide

Solution

3 4.

10 5

3 4 3 5

10 5 10 4 Write an equivalent multiplication.

3 5

5 2 2 2

Divide out the common factor, 5.

3

8

Because we are dividing two numbers that have different signs, the result is negative.


Dividing Decimal NumbersTo divide decimal numbers, set up a long division

and consider the divisor.Case 1: If the divisor is an integer, divide as if the

dividend were a whole number and place the decimal point in the quotient directly above its position in the dividend.

Case 2: If the divisor is a decimal number, 1. Move the decimal point in the divisor to the

right enough places to make the divisor an integer.

2. Move the decimal point in the dividend the same number of places.


Dividing Decimal Numbers continued

3. Divide the divisor into the dividend as if both numbers were whole numbers. Make sure you align the digits in the quotient properly.

4. Write the decimal point in the quotient directly above its new position in the dividend.

In either case, continue the division process until you get a remainder of 0 or a repeating digit (or block of digits) in the quotient.


Example 10

Divide 44.64 ÷ (3.6)

Solution Because the divisor is a decimal number, we move the decimal point enough places to the right to create an integer—in this case, one place. Then we move the decimal point one place to the right in the dividend. Because we are dividing two numbers with the same sign, the result is positive.


Example 10 continued

Divide 44.64 ÷ (3.6)

Solution 12.436 446.4

36

86

72

144

144

0


Example 11

Martha was decorating cookies. She used 2/3 of a container of frosting that was 3/4 full. What fractional part of the container did she use?

Solution To find 2/3 of 3/4, multiply.

2 3 2 3 1=

3 4 2 2 3 2


Multiply (–6)(–3)(7).

a) 126

b) 126

c) –63

d) 63

1.4


Divide

a)

b)

c)

d)

14.6 0.03 .

486.6

48.6

48.6

486.6

1.4


Exponents, Roots, and Order of Operations1.51.5

1. Evaluate numbers in exponential form.2. Evaluate square roots.3. Use the order-of-operations agreement to simplify numerical expressions.4. Find the mean of a set of data.


Objective 1

Evaluate numbers in exponential form.


Sometimes problems involve repeatedly multiplying the same number. In such problems, we can use an exponent to indicate that a base number is repeatedly multiplied.

Exponent: A symbol written to the upper right of a base number that indicates how many times to use the base as a factor.

Base: The number that is repeatedly multiplied.


When we write a number with an exponent, we say the expression is in exponential form. The expression is in exponential form, where the base is 2 and the exponent is 4. To evaluate , write 2 as a factor 4 times, then multiply.

4242

Exponent

42 2 2 2 2 = 16

Base

Four 2s


Evaluating an Exponential Form

To evaluate an exponential form raised to a natural number exponent, write the base as a factor the number of times indicated by the exponent; then multiply.


Example 1a

Evaluate. (–9)2

SolutionThe exponent 2 indicates we have two factors of –9. Because we multiply two negative numbers, the result is positive.

(–9)2 = (–9)(–9) = 81


Example 1b

Evaluate.

SolutionThe exponent 3 means we must multiply the base by itself three times.

33

5

33

5

3 3 3

5 5 5

27

125


Evaluating Exponential Forms with Negative Bases

If the base of an exponential form is a negative number and the exponent is even, then the product is positive.

If the base is a negative number and the exponent is odd, then the product is negative.


Example 2

Evaluate.a. b. c. d.Solutiona.

b.

c.

d.

4( 3)

4( 3) ( 3)( 3)( 3)( 3) 81

43 3( 2) 32

43 3 3 3 3 81

3( 2) ( 2)( 2)( 2) 8

32 2 2 2 8


Objective 2

Evaluate square roots.


Roots are inverses of exponents. More specifically, a square root is the inverse of a square, so a square root of a given number is a number that, when squared, equals the given number.

Square RootsEvery positive number has two square roots, a

positive root and a negative root. Negative numbers have no real-number square

roots.


Example 3

Find all square roots of the given number.

Solutiona. 49Answer 7

b. 81Answer No real-number square roots exist.


The symbol, called the radical, is used to indicate finding only the positive (or principal) square root of a given number. The given number or expression inside the radical is called the radicand.

,

25 5

Radicand

RadicalPrincipal Square Root


Square Roots Involving the Radical Sign

The radical symbol denotes only the positive (principal) square root.

, where 0 and 0. a a

a bb b


Example 4

Evaluate the square root.a. b. c. d.

Solutiona.

c.

169 250.64

169 13

0.64 0.8

64

81

64 8b.

81 9

d. 25 not a real number


Objective 3

Use the order-of-operations agreement to simplify numerical expressions.


Order-of- Operations Agreement

Perform operations in the following order:1. Within grouping symbols: parentheses ( ),

brackets [ ], braces { }, above/below fraction bars, absolute value | |, and radicals .

2. Exponents/Roots from left to right, in order as they occur.

3. Multiplication/Division from left to right, in order as they occur.

4. Addition/Subtraction from left to right, in order as they occur.


Example 5a

Simplify.

Solution

26 15 ( 5) 2

26 15 ( 5) 2

26 ( 3) 2

26 ( 6)

32

Divide 15 ÷ (5) = –3

Multiply (–3) 2 = –6

Add –26 + (–6) = –32


Example 5b

Simplify.

Solution

43 2 12 20

43 2 12 20 43 2 8

43 2 8

81 2 8

Subtract inside the absolute value.

Simplify the absolute value.

Evaluate the exponent.

81 16

65

Multiply.

Add.


Example 5c

Simplify.

Solution

23 5 6 2 1 49

23 5 6 2 1 49

23 5 6 3 49

9 5 3 7

9 15 7

24 7

17

Calculate within the innermost parenthesis.

Evaluate the exponential form, brackets, and square root.

Multiply 5(3).

Add 9 + 15.

Subtract 24 – 7.


Square Root of a Product or QuotientIf a square root contains multiplication or division, we can multiply or divide first, then find the square root of the result, or we can find the square roots of the individual numbers, then multiply or divide the square roots.

Square Root of a Sum or DifferenceWhen a radical contains addition or subtraction, we must add or subtract first, then find the root of the sum or difference.


Example 6a

Simplify.

Solution

213.5 5 4 3 142 21

10.2

Subtract within the radical.

Evaluate the exponential form and root.

Divide.

Multiply.

Subtract.

213.5 5 4 3 142 21

213.5 5 4 3 121

13.5 5 16 3(11)

2.7 16 3 11

43.2 33


Sometimes fraction lines are used as grouping symbols. When they are, we simplify the numerator and denominator separately, then divide the results.


Example 7a

Simplify.

Solution

38( 5) 2

4(8) 8

Evaluate the exponential form in the numerator and multiply in the denominator.

Multiply in the numerator and subtract in the denominator.

Subtract in the numerator.

Divide.

38( 5) 2

4(8) 8

8( 5) 8

4(8) 8

40 8

32 8

48

24

2


Example 7b

Simplify.

Solution

3

9(4) 12

4 (8)( 8)

Because the denominator or divisor is 0, the answer is undefined.

3

9(4) 12

4 (8)( 8)

36 12

64 (8)( 8)

48

64 ( 64)

48

0


Objective 4

Find the mean of a set of data.


Finding the Arithmetic MeanTo find the arithmetic mean, or average, of n numbers, divide the sum of the numbers by n.

Arithmetic mean = 1 2 ... nx x x

n


Example 8

Bruce has the following test scores in his biology class: 92, 96, 81, 89, 95, 93. Find the average of his test scores.

Solution

92 96 81 89 95 93

6

546

6

91

Divide the sum of the 6 scores by 6.


Simplify using order of operations.

a) 18

b) 6

c) 30

d) 36

26 18 9 6

1.5



a) 18

b) 6

c) 30

d) 36

26 18 9 6

1.5



a)

b)

c)

d) undefined

3

2

2 4 2

6 30 2 4

8

300

250

361

2

11

1.5


Translating Word Phrases to Expressions1.61.6

1. Translate word phrases to expressions.


Objective 1

Translating word phrases to Expressions


Translating Basic PhrasesAddition Translation Subtraction Translation

The sum of x and three

x + 3 The difference of x and three

x – 3

h plus k h + k h minus k h – k

seven added to t 7 + t seven subtracted from t

t – 7

three more than a number

n + 3 three less than a number

n – 3

y increased by two

y + 2 y decreased by two y – 2

Note: Since addition is a commutative operation, it does not matter in what order we write the translation.

Note: Subtraction is not a commutative operation; therefore, the way we write the translation matters.


Translating Basic PhrasesMultiplication Translation Division Translation

The product of x and three

3x The quotient of x and three

x 3 or

h times k hk h divided by k h k or

Twice a number 2n h divided into k k h or

Triple the number

3n The ratio of a to b a b or

Two-thirds of a number

Note: Like addition, multiplication is a commutative operation: it does not matter in what order we write the translation.

Note: Division is like subtraction in that it is not a commutative operation; therefore, the way we write the translation matters.

2

3n

3

x

h

kk

ha

b


Translating Basic PhrasesExponents Translation Roots Translation

c squared c2 The square root of x

The square of b b2

k cubed k3

The cube of b b3

n to the fourth power

n4

y raised to the fifth power

y5

x


The key words sum, difference, product, and quotient indicate the answer for their respective operations.

sum of x and 3

x + 3

difference of x and 3

product of x and 3 quotient of x and 3

x – 3

x 3 x 3


Example 1

Translate to an algebraic expression.a. five more than two times a numberTranslation: 5 + 2n or 2n + 5

b. seven less than the cube of a numberTranslation: n3 – 7

c. the sum of h raised to the fourth power and twelveTranslation: h4 + 12


Translating Phrases Involving Parentheses

Sometimes the word phrases imply an order of operations that would require us to use parentheses in the translation.

These situations arise when the phrase indicates that a sum or difference is to be calculated before performing a higher-order operation such as multiplication, division, exponent, or root.


Example 2

Translate to an algebraic expression.a. seven times the sum of a and bTranslation: 7(a + b)

b. the product of a and b divided by the sum of w2 and 4

Translation: ab (w2 + 4) or 2 4

ab

w


Translate the phrase to an algebraic expression. Twelve less than three times a number

a) 3n + 12

b) 12 – 3n

c) 3n – 12

d) 3n 12

1.6


Translate the phrase to an algebraic expression.

Twelve less than three times a number

a) 3n + 12

b) 12 – 3n

c) 3n – 12

d) 3n 12

1.6



The difference of a and b decreased by the sum of w and z

a) (a – b) – (w + z)

b) a – b – (w + z)

c) ab – (w + z)

d) (b – a) – (w + z)

1.6


Evaluating and Rewriting Expressions1.71.7

1. Evaluate an expression.2. Determine all values that cause an expression to be undefined.3. Rewrite an expression using the distributive property.4. Rewrite an expression by combining like terms.


Objective 1

Evaluate an expression.


Evaluating an Algebraic Expression

To evaluate an algebraic expression:1. Replace the variables with their corresponding

given values.2. Calculate the numerical expression using the order

of operations.


Example 1a

Evaluate 3w – 4(a – 6) when w = 5 and a = 7.

Solution3w – 4(a 6)

3(5) – 4(7 – 6)= 3(5) – 4(1)= 15 – 4= 11

Replace w with 5 and a with 7.

Simplify inside the parentheses first.

Multiply.

Subtract.


Example 1b

Evaluate x2 – 0.4xy + 9, when x = 7 and y = –2.

Solutionx2 – 0.4xy + 9

(7)2 – 0.4(7)(–2) + 9= 49 – 0.4(7)(–2) + 9= 49 – (–5.6) + 9= 49 + 5.6 + 9= 63.6

Replace x with 7 and y with –2.

Begin calculating by simplifying the exponential form.

Multiply.

Write the subtraction as an equivalent addition.

Add from left to right.


Objective 2

Determine all values that cause an expression to be undefined.


When evaluating a division expression in which the divisor or denominator contains a variable or variables, we must be careful about what values replace the variable(s).

We often need to know what values could replace the variable(s) and cause the expression to be undefined or indeterminate.


Example 2Determine all values that cause the expression to be undefined.a. b.

Answera. If x = 4, we have which is undefined because the denominator is 0.b. If x = 2 or 9 the fraction will be undefined since the denominator will = 0.

8

4x 2

( 2)( 9)x x

8 8,

4 4 0

2 2

( 2 2)( 2 9) 0

2 2

(9 2)(9 9) 0


Objective 3

Rewrite an expression using the distributive property.


The Distributive Property of Multiplication over Addition

a(b + c) = ab + ac

This property gives us an alternative to the order of operations.

2(5 + 6) = 2(11) 2(5 + 6) = 25 + 26

= 22 = 10 + 12

= 22


Example 3

Use the distributive property to write an equivalent expression and simplify.a. 3(x + 3) b. –5(w – 4)

Solutiona. 3(x + 3) = 3 x + 3 3

= 3x + 9

b. –5(w – 4) = –5 w – (–5) 4= –5w + 20


Objective 4

Rewrite an expression by combining like terms.


Terms: Expressions that are the addends in an expression that is a sum.

Coefficient: The numerical factor in a term.The coefficient of 5x3 is 5.The coefficient of –8y is –8.

Like terms: Variable terms that have the same variable(s) raised to the same exponents, or constant terms.Like terms Unlike terms4x and 7x 2x and 8y different variables

5y2 and 10y2 7t3 and 3t2 different exponents

8xy and 12xy x2y and xy2 different exponents

7 and 15 13 and 15x different variables


Combining Like TermsTo combine like terms, add or subtract the coefficients and keep the variables and their exponents the same.


Example 4

Combine like terms.a. 10y + 8y

Solution 10y + 8y = 18y

b. 8x – 3x Solution 8x – 3x = 5x

c. 13y2 – y2

Solution 13y2 – y2 = 12y2


Example 5

Combine like terms in 5y2 + 6 + 4y2 – 7.

Solution 5y2 + 6 + 4y2 – 7= 5y2 + 4y2 + 6 – 7 Combine like terms.

= 9y2 – 1


Example 6

Combine like terms in 18y + 7x – y – 7x.

Solution 18y + 7x – y – 7x

= 17y + 0

= 17y


Example 7

Combine like terms inSolution

1 14 3 .

12 6a b a b

1 14 3

12 6a b a b

1 14 3

12 6a a b b Collect like terms.

Write the fraction coefficients as equivalent fractions with their LCD, 12.

1 1(2)4 3

12 6(2)a a b b

1 24 3

12 12a a b b

33 3

12a b

13 3

4a b Combine like terms.


Evaluate the expression 4(a + b) when a = 3 and b = –2.

a) 4

b) 4

c) 12

d) 20

1.7


For which values is the expression undefined?

a) 8

b) 2

c) 2 and 5

d) 2 and 5

8

( 2)( 5)

m

m m

1.7


Simplify: 7x + 8 – 2x – 4

a) 9x – 4

b) 9x + 4

c) 5x – 4

d) 5x + 4

1.7

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