slide 7- 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Real Numbers and Algebraic Expressions

7.1 Introduction to Algebra7.2 The Real Numbers7.3 Addition of Real Numbers7.4 Subtraction of Real Numbers7.5 Multiplication of Real Numbers7.6 Division of Real Numbers7.7 Properties of Real Numbers7.8 Simplifying Expressions; Order of Operations

77


INTRODUCTION TO ALGEBRA

Evaluate algebraic expressions by substitution.

Translate phrases to algebraic expressions.

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Objective

Evaluate algebraic expressions by substitution.

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Algebraic Expressions

An algebraic expression consists of variables, constants, numerals, and operation signs.

x + 38 19 – y

When we replace a variable with a number, we say that we are substituting for the variable.

This process is called evaluating the expression.

5a x

y


Example A

Evaluate x + y for x = 38 and y = 62.

SolutionWe substitute 38 for x and 62 for y.

x + y = 38 + 62 = 100


Example B

Evaluate and for x = 72 and y = 8.

SolutionWe substitute 72 for x and 8 for y:

x

y

x

y

89

72

x

y

729

8

x

y


Example C

Evaluate for C = 30.

SolutionThis expression can be used to find the Fahrenheit temperature that corresponds to 30 degrees Celsius.

932

5

C

932

5

C 30932

5

270

325

54 32 86.


Objective

Translate phrases to algebraic expressions.

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Translating to Algebraic Expressions

per of decreased by increased by

ratio of twice less than more than

divided into times minus plus

quotient product difference sum

divided bymultiplied bysubtracted from added to

DivisionMultiplicationSubtractionAddition


Example DTranslate each phrase to an algebraic expression.a) 9 more than yb) 7 less than xc) the product of 3 and twice w

SolutionPhrase Algebraic Expression

a) 9 more than y y + 9b) 7 less than x x 7c) the product of 3 and twice w 3•2w or 2w • 3


Example E

Translate each phrase to an algebraic expression.

Phrase Algebraic Expression

Eight more than some number

One-fourth of a number

Two more than four times some number

Eight less than some number

Five less than the product of two numbers

Twenty-five percent of some number

Seven less than three times some number

x + 8, or 8 + x

4x + 2, or 2 + 4x

1, , or / 4

4 4

xx x

n – 8

ab – 5

0.25n

3w – 7


THE REAL NUMBERS

State the integer that corresponds to a real-world situation.Graph rational numbers on a number line.Convert from fraction notation to decimal notation for a rational number.Determine which of two real numbers is greater and indicate which, using < or >; given an inequality like a > b, write another inequality with the same meaning. Determine whether an inequality like 3 5 is true or false.Find the absolute value of a real number.

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Objective

State the integer that corresponds to a real-world situation.

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Natural NumbersThe set of natural numbers = {1, 2, 3, …}. These are the numbers used for counting.

Whole NumbersThe set of whole numbers = {0, 1, 2, 3, …}. This is the set of natural numbers with 0 included.


IntegersThe set of integers = {…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, …}.


IntegersIntegers consist of the whole numbers and their opposites.

Integers to the left of zero on the number line are called negative integers and those to the right of zero are called positive integers. Zero is neither positive nor negative and serves as its own opposite.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0, neither positive nor negative

Positive integersNegative integers

Opposites


Example ATell which integer corresponds to each situation.1. Death Valley is 282 feet below sea level.2. Margaret owes $312 on her credit card. She has

$520 in her checking account.

Solution

1. 282 below sea level corresponds to 282.

2. The integers 312 and 520 correspond to the situation.


Objective

Graph rational numbers on a number line.

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Fractions such as ½ are not integers. A larger system called rational numbers contains integers and fractions. The rational numbers consist of quotients of integers with nonzero divisors.The following are examples of rational numbers:

2 2 7 23, , , 4, 3, 0, , 2.4, 0.17

3 3 1 8


Rational NumbersThe set of rational numbers consist of all numbers that can be named in the form , where a and b are integers and b is not equal to 0 (b 0).

a

b


Example

To graph a number means to find and mark its point on the number line.

Graph:

Solution The number can be named , or 3.5. Its graph is halfway between 3 and 4.

7

27

2

13

2

10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102

3.5


Example CGraph: 2.8

Solution The graph of 2.8 is 8/10 of the way from 2 to 3.

10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102

-2.8


Objective

Convert from fraction notation to decimal notation for a rational number.

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Each rational number can be named using fraction notation or decimal notation.


Example D

Find decimal notation for

Solution Because means 7 40, we divide.

7.

40

7

400.175

40 7.000

40

300

280

200

200

0We are finished when the remainder is 0.

17

400. 75


Example EFind decimal notation for

Solution Divide 1 12

1.

12

0.083312 1.0000

0

100

96

0

36

0

36

4

4

4

Since 4 keeps reappearing as a remainder, the digits repeat and will continue to do so; therefore,

10.08333...

12 1

or 0.083.12


Objective

Determine which of two real numbers is greater and indicate which, using < or >; given an inequality like a > b, write another inequality with the same meaning. Determine whether an inequality like 3 5 is true or false.

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The Real-Number SystemThe set of real numbers = The set of all numbers corresponding to points on the number line.

Decimal notation for rational numbers either terminates or repeats.Decimal notation for irrational numbers neither terminates nor repeats.


Positive Integers: 1, 2, 3,

…

Integers Zero: 0

Rational numbers

Negative integers: -1, -2, -

3, …

Real numbers

Rational numbers that are not integers: 2/3, -4/5, 19/-5, -7/8, 8.2,

Irrational numbers: pi, 5.363663666…


Numbers are written in order on the number line, increasing as we move to the right. For any two numbers on the line, the one to the left is less than the one to the right.

The symbol < means “is less than,” 4 < 8 is read “4 is less than 8.”

The symbol > means “is greater than,” 6 > 9 is read “6 is greater than 9.”


Example FUse either < or > for to form a true sentence.1. 7 3 2. 8 3 3. 21 9

Solution

10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102

1. 7 3 3. 21 92. 8 3

Since 7 is to the left of 3, we have 7 < 3. Since 8 is to the right of

3, we have 8 > 3.

Since 21 is to the left of 9, we have 21 < 9.

7 < 3. 8 > 3. 21 < 9


Example GUse either < or > for to form a true sentence.1. 7.2 2.

Solution1. 7.2

2. Convert to decimal notation:

5 8

8 13

5

2

5

2 10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102<

5 8

8 135

0.62588

0.615413

>


Order; >, <a < b also has the meaning b > a.


Example H

Write another inequality with the same meaning.a. 4 > 10b. c < 7

Solutiona. The inequality 10 < 4 has the same meaning.

b. The inequality 7 > c has the same meaning.


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Positive numbers bNegative numbers a

a < 0 b > 0

If b is a positive real number, then b > 0.If a is a negative real number, then a < 0.


Objective

Find the absolute value of a real number.

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Absolute ValueThe absolute value of a number is its distance from zero on a number line. We use the symbol |x| to represent the absolute value of a number x.

5 units from 0 5 units from 0


Finding Absolute Valuea) If a number is negative, its absolute value is

positive.b) If a number is positive or zero, its absolute value

is the same as the number.


Example IFind the absolute value of each number.

a. |5| b. |36|

c. |0| d. |52|

Solution

a. |5| The distance of 5 from 0 is 5, so | 5| = 5.

b. |36| The distance of 36 from 0 is 36, so |36| = 36.

c. |0| The distance of 0 from 0 is 0, so |0| = 0.

d. |52| The distance of 52 from 0 is 52, so |52| = 52.


ADDITION of REAL NUMBERS

Add real numbers without using a number line.

Find the opposite, or additive inverse, of a real number.

Solve applied problems involving addition of real numbers.

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Objective

Add real numbers without using a number line.

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Addition on a Number LineTo do the addition a + b, on a number line we start at 0. Then we move to a, and then move according to b.a) If b is positive, we move from a to the right.b) If b is negative, we move from a to the left.c) If b is 0, we stay at a.


Example A

Add: 3 + (6).

Solution

3 + (6) = 3

Start at 3.

Move

6 units to the left.


Example B

Add: 5 + 8.

Solution

5 + 8 = 3

Start at 5.Move

8 units to the right.


Rules for Addition of Real Numbers

1. Positive numbers: Add the same as arithmetic numbers. The answer is positive.

2. Negative numbers: Add absolute values. The answer is negative.

3. A positive and a negative number: Subtract the smaller absolute value from the larger. Then: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.


Example C

Add.1. 5 + (8) = 2. 9 + (7) =

Solution1. 5 + (8) = 13

2. 9 + (7) = 16

Add the absolute values: 5 + 8 = 13. Make the answer negative.


Example DAdd.

1. 4 + (6) = 2. 12 + (9) =

3. 8 + 5 = 4. 7 + 5 =

Solution

1. 4 + (6) =

2. 12 + (9) =

3. 8 + 5 =

4. 7 + 5 =

Think: The absolute values are 4 and 6. The difference is 2. Since the negative number has the larger absolute value, the answer is negative, 2.

2

3

32


Example EAdd: 16 + (2) + 8 + 15 + (6) + (14).

SolutionBecause of the commutative and associate laws for addition, we can group the positive numbers together and the negative numbers together and add them separately. Then we add the two results.

16 + (2) + 8 + 15 + (6) + (14)

= 16 + 8 + 15 + (2) + (6) + (14)

= 39 + (22)

= 17


Objective

Find the opposite, or additive inverse, of a real number.

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Opposites, or Additive InversesTwo numbers whose sum is 0 are called opposites, or additive inverses, of each other.


Example F

Find the opposite, or additive inverse, of each number.1. 52 2. 12 3. 0 4.

Solution1. 52 The opposite of 52 is 52 because 52 + (52) =

0

2. 12 The opposite of 12 is 12 because 12 + 12 = 0

3. 0 The opposite of 0 is 0 because 0 + 0 = 0

4. The opposite of is because

4

5

4

5

4

5

4

5 5

40

5

4


Symbolizing OppositesThe opposite, or additive inverse, of a number a can be named a (read “the opposite of a,” or “the additive inverse of a”).

The Opposite of an OppositeThe opposite of the opposite of a number is the number itself. (The additive inverse of the additive inverse of a number is the number itself.) That is, for any number a

(a) = a.


Example G

Evaluate x and (x) when x = 12.

SolutionWe replace x in each case with 12.a) If x = 12, then x = 12 = 12

b) If x = 12, then (x) = (12) = 12


Example HEvaluate (x) for x = 7.

SolutionWe replace x with 7.

If x = 7, then (x) = ((7)) = 7


The Sum of Opposites For any real number a, the opposite, or additive inverse, of a, expressed as a, is such that

a + (a) = a + a = 0.


Example I

Change the sign (Find the opposite.)a) 9 b) 8

Solutiona) 9 (9) = 9

b) 8 (8) = 8


Example JOn a recent day, the price of a stock opened at a value of $72.37. During the day, it rose $3.25, dropped $6.32, and rose $4.12. Find the value of the stock at the end of the day.Solution We let t = the stock price at the end of the dayt = starting price + 1st rise – drop + 2nd rise = $72.37 + $3.25 – $6.32 + $4.12 = $73.42The value of the stock at the end of the day was $73.42


SUBTRACTION of REAL NUMBERS

Subtract real numbers and simplify combinations of additions and subtractions.

Solve applied problems involving subtraction of real numbers.

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Objective

Subtract real numbers and simplify combinations of additions and subtractions.

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Subtraction a bThe difference a b is the number c for which a = b + c.


Example ASubtract 4 9.

Solution

Think: 4 9 is the number that when added to 9 gives 4.

What number can we add to 9 to get 4?

The number must be negative.

The number is 5:

4 – 9 = –5.

That is, 4 9 = 5 because 9 + (5) = 4.


Subtracting by Adding the Opposite

For any real numbers a and b, a – b = a + (– b).

(To subtract, add the opposite, or additive inverse, of the number being subtracted.)


Example B

Subtract.1. 15 (25) 2. 13 40

Solution

1. 15 (25) = 15 + 25 Adding the opposite of 25 = 10

2. 13 40 = 13 + (40) Adding the opposite of 40= 53


Example CSubtract.1. 3 – 7 = 2. –5 – 9 3. –4 – (–10)

Solution1. 3 – 7 = 3 + (–7)

= –4

2. –5 – 9 = –5 + (– 9) = –14

3. –4 – (–10) = –4 + 10 = 6

The opposite of 7 is –7. We change the subtraction to addition and add the opposite. Instead of subtracting 7, we add –7.


Example DSimplify: 4 (6) 10 + 5 (7).

Solution

4 (6) 10 + 5 (7) = 4 + 6 + (10) + 5 + 7

= 4 + (10) + 6 + 5 + 7

= 14 + 18

= 4Adding opposites

Using a commutative law


Objective

Solve applied problems involving subtraction of real numbers.

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Example E

The Johnson’s were taking a vacation and one day they drove from mile marker 54 to mile marker 376. How far did they drive?

Solution376 – 54 = 376 + (54)

= 322 miles

Adding the opposite of 54.


MULTIPLICATION of REAL NUMBERS

Multiply real numbers.

Solve applied problems involving multiplication of real numbers.

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Objective

Multiply real numbers.

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Multiplication of real numbers is like multiplication of arithmetic numbers. The only difference is that we must determine whether the answer is positive or negative.

The Product of a Positive and a Negative Number

To multiply a positive number and a negative number, multiply their absolute values. The answer is negative.


Example AMultiply.1. (7)(9) 2. 40(1) 3. 3 7

Solution

1. (7)(9) =

2. 40(1) =

3. 3 7 =

63

40

21


The Product of Two Negative Numbers

To multiply two negative numbers, multiply their absolute values. The answer is positive.


Example B

Multiply.1. (3)(4) 2. (11)(5) 3. (2)(1)

Solution1. (3)(4) =

2. (11)(5) =

3. (2)(1) =

12

55

2


To multiply two nonzero real numbers:

a) Multiply the absolute values.

b) If the signs are the same, the answer is positive.

c) If the signs are different, the answer is negative.


The Multiplication Property of Zero

For any real number a, a 0 = 0 a = 0.

(The product of 0 and any real number is 0.)


Example CMultiply.1. 9 3(4) 2. 6 (3) (4) (7)

Solution1. 9 3(4) = 27(4)

= 108

2. 6 (3) (4) (7) = 18 28 = 504

Multiplying the first two numbers

Multiplying the results

Each pair of negatives gives a positive product.


The product of an even number of negative numbers is positive.

The product of an odd number of negative numbers is negative.


Example D

Evaluate 3x2 when x = 4 and x = 4.

Solution3x2 = 3(4)2

= 3(16) = 48

3x2 = 3(4)2

= 3(16)

= 48


Example E

Evaluate (x)2 and x2 when x = 6.

Solution(x)2 = (6)2

= (6)(6) = 36

x2 = (6)2

= 36


Objective

Solve applied problems involving multiplication of real numbers.

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Example F

The temperature in a chemical compound was 4 at 1:00. During a reaction, it increased in temperature 2 per minute until 1:12. What was the temperature at 1:12?SolutionNumber of minutes temp. rises is 12. (1:12 – 1:00)Rises 2 per minute = 2(12) = 24 4 + 24 = 28 The temperature was 28 at 1:12.


DIVISION of REAL NUMBERS

Divide integers.

Find the reciprocal of a real number.

Divide real numbers.

Solve applied problems involving division of real numbers.

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Objective

Divide integers.

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Division

The quotient a b or , where b 0, is that unique real number c for which a = b c.

a

b


Example ADivide, if possible. Check each answer.

1. 15 (3) 2.

Solution1. 15 (3) = 5

2.

45

5

459

5

Think: What number multiplied by –3 gives 15? The number is –5. Check: (–3)(–5) = 15.

Think: What number multiplied by –5 gives 45? The number is –9. Check: (–5)(–9) = 45.


To multiply or divide two real numbers (where the divisor is nonzero):a) Multiply or divide the absolute values.b) If the signs are the same, the answer is positive.c) If the signs are different, the answer is negative.

Excluding Division by 0Division by zero is not defined: a 0, or is not defined for all real numbers a.

,0

a


Dividends of 0Zero divided by any nonzero real number is 0:

00, 0. a

a


Example BDivide, if possible: 72 0.

Solution

is undefined.72

0

Think: What number multiplied by 0 gives 72? There is no such number because anything times 0 is 0.


Objective

Find the reciprocal of a real number.

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ReciprocalsTwo numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other.


Example CFind the reciprocal.

1. 2. 6 3. 4.

Solution1. The reciprocal of is

2. The reciprocal of 6 is

3. The reciprocal of is

4. The reciprocal of is

6

7

1

4

8

9

6

7

1

48

9

7

6.

.1

6

4.

.9

8


Reciprocal PropertiesFor a 0, the reciprocal of a can be named and the reciprocal of is a.The reciprocal of any nonzero real number can be named The number 0 has no reciprocal.

1

a1

a

a

b.

b

a

The Sign of a ReciprocalThe reciprocal of a number has the same sign as the number itself.


NumberOpposite

(Change the sign.)

Reciprocal (Invert but do not change the sign.)

25 25

8.5 8.5

0 0 Undefined

3

4

3

4

4

3

1

2523

3

23

3

3

231 10

or 8.5 85


Objective

Divide real numbers.

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Reciprocals and DivisionFor any real numbers a and b, b 0,

(To divide, multiply by the reciprocal of the divisor.)

1

aa b a

b b


Example D

Rewrite the division as multiplication.1. 9 5 2.

Solution1. 9 5 is the same as

2.

3 2

4 5

19

5

3 2

4 5

3 5

4 2


Example EDivide by multiplying by the reciprocal of the divisor. Solution

3 5

4 16

3 5

4 16

63

4

1

5

4

3 44

5

44

4

3

5

12

5

Multiply by the reciprocal of the divisor

Factoring and identifying a common factor

Removing a factor equal to 1


Example F

Divide. 18.6 (3)

Solution18.6 (3) =

18.66.2

3

Do the long division. The

answer is negative.

6.23 18.6


Objective

Solve applied problems involving division of real numbers.

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Example GAfter diving 110 m below sea level, a diver rises at a rate of 8 meters per minutes for 6 minutes. Where is the diver in relation to the surface?SolutionWe first determine by how many meters the diver rose altogether.8 meters 6 = 48 metersThe diver was 110 below sea level and rose 48 meters.110 + 48 = 62 meters or 62 meters below sea level.

110 m


PROPERTIES OF REAL NUMBERS

Find equivalent fraction expressions and simplify fraction expressions.Use the commutative and associative laws to find equivalent expressions.Use the distributive laws to multiply expressions like 8 and x – y.Use the distributive laws to factor expressions like 4x – 12 + 24y.Collect like terms.

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Objective

Find equivalent fraction expressions and simplify fraction expressions.

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Equivalent ExpressionsTwo expressions that have the same value for all allowable replacements are called equivalent.

The Identity Property of 0For any real number a,

a + 0 = 0 + a = a(The number 0 is the additive identity.)


The Identity Property of 1For any real number a,

a 1 = 1 a = a(The number 1 is the multiplicative identity.)


Example A

Simplify:

Solution

40.

24

x

x

40 5 8

24 3 8

x x

x x5 8

3 8

x

x5

13

5

3

Look for the largest factor common to both the numerator and the denominator and factor each.Factoring the fraction expression

8x/8x = 1

Removing a factor of 1 using the identity property of 1


Objective

Use the commutative and associative laws to find equivalent expressions.

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Example B

Evaluate x + y and y + x when x = 7 and y = 8.

SolutionWe substitute 7 for x and 8 for y.x + y = 7 + 8 = 15

y + x = 8 + 7 = 15


Example C

Evaluate xy and yx when x = 7 and y = 5.

SolutionWe substitute 7 for x and 5 for y.xy = 7(5) = 35

yx = 5(7) = 35


The Commutative Laws Addition: For any numbers a, and b,

a + b = b + a.(We can change the order when adding without affecting the answer.)Multiplication. For any numbers a and b,

ab = ba(We can change the order when multiplying without affecting the answer.)


Example D

Calculate and compare:4 + (9 + 6) and (4 + 9) + 6.

Solution4 + (9 + 6) = 4 + 15 = 19

(4 + 9) + 6 = 13 + 6 = 19


The Associative Laws Addition: For any numbers a, b, and c,

a + (b + c) = (a + b) + c.(Numbers can be grouped in any manner for addition.)Multiplication. For any numbers a, b, and c,

a (b c) = (a b) c(Numbers can be grouped in any manner for multiplication.)


Objective

Use the distributive laws to multiply expressions like 8 and x – y.

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The Distributive Law of Multiplication over Addition For any numbers a, b, and c,

a(b + c) = ab + ac.

The Distributive Law of Multiplication over SubtractionFor any numbers a, b, and c,

a(b c) = ab ac.


Example EMultiply. 4(a + b).

Solution

4(a + b) = 4 a + 4 b

= 4a + 4b

Using the distributive law of multiplication over addition


Example F

Multiply. 1. 8(a – b) 2. (b – 7)c 3. –5(x – 3y + 2z)

Solution1. 8(a – b) = 8a – 8b

2. (b – 7)c = c(b – 7) = c b – c 7 = cb – 7c


continued

3. –5(x – 3y + 2z) = –5 x – (–5 3)y + (–5 2)z

= –5x – (–15)y + (–10)z

= –5x + 15y – 10z


Objective

Use the distributive laws to factor expressions like 4x – 12 + 24y.

dd


Factoring is the reverse of multiplying. To factor, we can use the distributive laws in reverse:ab + ac = a(b + c) and ab – ac = a(b – c).

FactoringTo factor an expression is to find an equivalent expression that is a product.


Example G

Factor.a. 6x – 12 b. 8x + 32y – 8

Solutiona. 6x – 12 = 6 x – 6 2

= 6(x – 2)

b. 8x + 32y – 8 = 8 x + 8 4y – 8 1 = 8(x + 4y – 1)


Example H

Factor. Try to write just the answer, if you can.a. 7x – 7y b. 14z – 12x – 20

Solutiona. 7x – 7y =

b. 14z – 12x – 20 =

7(x – y)

2(7z – 6x – 10)


Objective

Collect like terms.

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A term is a number, a variable, a product of numbers and/or variables, or a quotient of two numbers and/or variables.

Terms are separated by addition signs. If there are subtraction signs, we can find an equivalent expression that uses addition signs.


Like TermsTerms in which the variable factors are exactly the same, such as 9x and –5x, are called like, or similar terms.

Like Terms Unlike Terms

7x and 8x 8y and 9y2

3xy and 9xy 5a2b and 4ab2

The process of collecting like terms is based on the distributive laws.


Example ICombine like terms. Try to write just the answer.1. 8x + 2x 2. 3x – 6x3. 3a + 5b + 2 + a – 8 – 5b

Solution1. 8x + 2x = (8 + 2) x

= 10x

3. 3a + 5b + 2 + a – 8 – 5b = 3a + 5b + 2 + a + (–8) + (–5b) = 3a + a + 5b + (–5b) + 2 + (–8) = 4a + (–6) = 4a – 6

2. 3 6 (3 6) x x x

3 x


SIMPLIFYING EXPRESSION; ORDER OF OPERATIONS

Find an equivalent expression for an opposite without parentheses, where an expression has several terms.Simplify expressions by removing parentheses and collecting like terms.Simplify expressions with parentheses inside parentheses.Simplify expressions using rules for order of operations.

7.7.88aa

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Objective

Find an equivalent expression for an opposite without parentheses, where an expression has several terms.

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The Property of 1For any real number a,

1 a = a(Negative one times a is the opposite, or additive inverse, of a.)


Example A

Find an equivalent expression without parentheses. (4x + 5y + 2)Solution(4x + 5y + 2) = 1(4x + 5y + 2)

= 1(4x) + 1(5y) + 1(2) = 4x – 5y – 2

Using the property of 1

Using a distributive law

Using the property of 1


Example B

Find an equivalent expression without parentheses. (2x + 7y 6)

Solution(2x + 7y 6) = 2x – 7y + 6 Changing the sign of each term


Objective

Simplify expressions by removing parentheses and collecting like terms.

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Example C

Remove parentheses and simplify. (8x + 5y – 3) (4x – 2y 6)

Solution

(8x + 5y – 3) (4x – 2y 6)= 8x + 5y – 3 – 4x + 2y + 6= 4x + 7y + 3


Example D

Remove parentheses and simplify.(3a + 4b – 8) – 3(–6a – 7b + 14)

Solution(3a + 4b – 8) – 3(–6a – 7b + 14)

= 3a + 4b – 8 + 18a + 21b – 42= 21a + 25b – 50


Objective

Simplify expressions with parentheses inside parentheses.

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When more than one kind of grouping symbol occurs, do the computations in the innermost ones first. Then work from the inside out.


Example E

Simplify. 5(3 + 4) – {8 – [5 – (9 + 6)]}

Solution 5(3 + 4) – {8 – [5 – (9 + 6)]}

= 5(7) – {8 – [5 – 15]}= 35 – {8 – [ –10]} Computing 5(7) and 5 –

15

= 35 – 18 Computing 8 – [–10] = 17


Example F

Simplify. [6(x + 3) – 4x] – [4(y + 3) – 8(y – 4)]Solution

= [6x + 18 – 4x] – [4y + 12 – 8y + 32]

= [2x + 18] – [4y + 44] Collecting like terms within brackets

= 2x + 18 + 4y – 44 Removing brackets

= 2x + 4y – 26 Collecting like terms


Objective

Simplify expressions using the rules for order of operations.

dd


Rules for Order of Operations1. Do all calculations within grouping symbols

before operations outside.2. Evaluate all exponential expressions.3. Do all multiplications and divisions in order from

left to right.4. Do all additions and subtractions in order from

left to right.


Example GSimplify.

1. 2.

Solution1.

20 12 4 2 3( 3) 9 6( 3)

20 12 4 2 20 2 02 24 321

620 14

3( 3) 9 6( 3) 3( 3) 9 6( 3) 2 9 6(7 3)

6(3 3) )3 6( 3

13 ( 8)

21

2.


Example H

Simplify:

Solution

6 3 9.

2

2

18 9

2

2

1

6 3 9

2

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