reteaching 6 subtracting real · pdf filethen follow the rules for adding real numbers. find...

Name Date Class

© Saxon. All rights reserved. 11 Saxon Algebra 1

You have added real numbers. Now you will subtract real numbers.Two numbers with the same absolute value but different signs are called opposites. The opposite of a number is also called the additive inverse. The sum of a number and its additive inverse is 0.

To Subtract Real Numbers

To subtract a number, add its inverse. Then follow the rules for adding real numbers.

Find the difference (�6) � 12.

Step 1: To subtract a number, add its inverse. (�6) � (�12)

Step 2: Add the absolute values. 6 � 12 � 18

Step 3: Use the sign of the numbers. The sign is negative.

(�6) � 12 � �18

Find the difference (�8) � (�15).

Step 1: To subtract a number, add its inverse. (�8) � 15

Step 2: Find the difference of the absolute values. 15 � 8 � 7

Step 2: Find the sign of the number with the largest absolute value. � 15 � � � �8 � , so the sign is positive.

(�8) � (�15) � 7

PracticeComplete the steps to find each difference.

1. (�6) � (�11)

(�6) � (�11) � (�6) � 11

11 � (6) � 5 � �11 � � � �6 � , so the sign of the answer

is positive .

(�6) � (�11) � 5

2. (�19) � 3

(�19) � 3 � (�19) � (�3)

19 � 3 � 22 The sign of the numbers is negative,

so the sign of the answer is negative.

(�19) � 3 � �22

Find each difference.

3. � �17 � � (�12) � �5 4. 25 � (�32) � 57

5. � � 5 __ 8 � � 1 __

4 � � 7 __

8 6. 5 __

6 � � � 1 ___

12 � � 11 ___

12

7. (�9.1) � 2.6 � �11.7 8. 7 � 26 � �19

9. For safety, scuba divers usually do not dive deeper than 40 meters below sea level. A diver in a helmet suit can safely dive about 21 meters deeper than a scuba diver. What is the maximum safe depth for a helmet suit diver in relation to sea level?

�61 m

6ReteachingSubtracting Real Numbers


Reteachingcontinued

Closure

A set is closed under a given operation if the outcome of the operation on any two members of the set is also a member of the set.

Determine whether the statement is true or false. Give a counterexample if the statement is false.

The set of natural numbers is closed under subtraction.

Subtract two natural numbers:

3 � 2 � 1

4 � 2 � 2

2 � 3 � �1

The last statement is a counterexample, so the statement is false.

PracticeComplete the steps to determine whether the statement is true or false. Write a counterexample if the statement is false.

10. The set A � {�1, 0, 1} is closed under subtraction.

Subtract two numbers in the set:

1 � 0 � 1

0 � 1 � �1

�1 � 1 � �2

The last statement is a counterexample so the statement is false.

Determine whether each statement is true or false. Give a counterexample for false statements.

11. The set of even integers, {..., �4, �2, 0, 2, 4, ...}, is closed under subtraction.

true

12. The set of irrational numbers is closed under subtraction.

false; sample counterexample: ��

2 � ��

2 � 0

13. The set of odd integers plus zero, {..., �5, �3, �1, 0, 1, 3, 5, ...} is closed under subtraction.

false; sample counterexample: 3 � 1 � 2

14. The set of integers that are a multiple of 4, {..., �12, �8, �4, 0, 4, 8, 12, ...} is closed under subtraction.

true

6

Name Date Class


You have used the order of operations to simplify expressions. Now you will apply this concept to simplifying expressions within symbols of inclusion.

Symbols of inclusion indicate which numbers, variables, and operations are parts of the same term. Some symbols of inclusion are fraction bars, absolute value symbols, parentheses, braces, and brackets.

Simplify 3[11 � (12 � 9)2] � 7. Justify each step.

3[11 � (12 � 9)2] � 7

� 3[11 � 32] � 7

� 3[11 � 9] � 7

� 3 � 20 � 7

� 60 � 7

� 67

PracticeComplete the steps to simplify the expression.

1. 6z � 4 _____ 3 � [2z � � 5 � 7 � ]

6z � 4 _____ 3

� [2z � � 5 � 7 � ]

6z � 4 _____ 3

� [2z � � �2 � ]

6z � 4 _____ 3

� [2z �2]

24z ____ 3

� 2z � 2

8z � 2z � 2

10z � 2

Simplify.

2. � 11 � 19 � � 15 � 23 3. 4(18 � 3) � (14 � 10 � � �6 � ) � 6

4. [ (9 � 2)2 � 2(13 � 4) ] � 3 � 34 5. 5 [ 24 � (11 � 9)3 ] � 4 � 20

6. 14 � 2 _____ 4 � 3(7 � 2)2 � 5 � 22 7. [ (16 � 7)2 � 3 � 7 _____

2 � 6 ] � 3 � 17

Simplify inside the parentheses.

Evaluate the exponent.

Add inside the brackets.

Multiply.

Add.

Simplify inside the parentheses.


Add inside the brackets.

Multiply.

Add.

Subtract inside absolute value symbols.

Simplify absolute value.

Simplify the numerator.

Simplify the fraction.

Add .

Subtract inside absolute value symbols.

Simplify absolute value.

Simplify the numerator.

Simplify the fraction.

Add .

ReteachingSimplifying and Comparing Expressions with Symbols of Inclusion 7


Reteachingcontinued

Compare the expressions. Use �, �, or �.

[ 21 � 3(9 � 5)2 ] � 50 � (12 � 7)2 � [ 19 � 3(21 � 15) ]

Simplify each expression. Then compare.

[ 21 � 3 (9 � 5) 2 ] � 50 (12 � 7) 2 � [ 19 � 3 (21 � 15) ] � (5) 2 � [ 19 � 3 (21 � 15) ] � 25 � [ 19 � 3 (21 � 15) ] � 25 � [ 19 � 3 (6) ] � 25 � [ 19 � 18 ] � 25 �1

� 24

� [ 21 � 3 (4) 2 ] � 50

� [ 21 � 3 � 16 ] � 50

� [ 21 � 48 ] � 50

� [ �27 ] � 50

� 50

Since 23 � 24, [ 21 � 3 (9 � 5) 2 ] � 50 �� (12 � 7) 2 � [ 19 � 3 (21 � 15) ] .

PracticeComplete the steps to simplify each expression. Compare the expressions. Use �, �, or �.

8. 2(17 � 8) � [ 3 2 �(8 � 6) ] � [ 3 � (�2) 3 ] � 4(5 � 2)

2 (17 � 8) � [ 3 2 � (8� 6) ] [ 3 � (�2) 3 ] � 4(5 � 2)

� [ 3 � ( �8 ) ] � 4(5 � 2)

� (�24) � 4(5 � 2)

� �24 � 4( 7 )

� �24 � 28� 4

� 2 (17 � 8) � [ 3 2 � 2 ] � 2 (17 � 8) � [ 9 � 2 ] � 2(17 � 8) � 7 � 2( 9 ) � 7 � 18 � 7 � 11

Since11 � 4, 2(17 � 8) � [ 3 2 � (8 � 6) ] �� [ 3 � (�2) 3 ] � 4(5 � 2).


9. 6 2 � [ 4(3 � 2) � 2 3 ] �� 5(9 � 3) � 4 [ 6 � (�2) 3 ]

10. 3 [ 6 2 � 5(22 � 2 4 ) � 1 ] �� [ (14 � 9) 2 � 6 ] � 2 3

11. 9 � [ 3 � 19 � 3 ______ 4 � � 5 ] � 4 2 �� 23 � 8 ______

3 � 2 � [ 2(6 � 3) + 3 2 ]

7

Name Date Class


You have used ratios to compare two quantities. Now you will use unit ratios to convert measures into different units. Unit analysis is the process of converting measures into different units.

The peregrine falcon can reach speeds up to 200 miles per hour. How fast is this in yards per hour?

Step 1: Identify the known and missing information.

200 mi ______ 1 hour

� ? yd

______ 1 hour

So, the conversion is 200 mi → ? yd.

Step 2: Equate units.

1 mi � 1,760 yd So, the unit ratio is 1760 yd

_______ 1 mi

, or 1mi _______ 1760 yd

.

Step 3: Write the multiplication sentence. Then multiply.

200 mi ______ 1 hr

� 1760 yd

________ 1 mi

200 mi ______ 1 hr

� 1760 yd

_______ 1 mi

Cancel out common factors.

� 200 � 1760 yd

____________ 1 hr

Multiply.

� 352,000 yd

__________ 1 hr

Write the ratio of yards per hour.

The peregrine falcon can reach speeds up to 352,000 yards per hour.

Practice

1. Alberto Contador won the 2007 Tour de France with an averagespeed of about 39 kilometers per hour. What was Alberto’s average speed in meters per hour?

39 km ______ 1hr

� 1000 m _________

1 km � 39 � 1000 m ___________

1 hr �

39,000 m ___________ 1 hr

Alberto Contador’s average speed was about 39,000 meters per hour.

2. Some elephants can eat up to 660 pounds of food per day. How much food can an elephant eat in tons per day? One ton is equal to 2000 pounds.

0.33 t/d

3. A sprinkler with a flow rate of 2 gallons per minute is watering a lawn. What is the flow rate of the sprinkler in gallons per hour?

120 gal/h

8ReteachingUsing Unit Analysis to Convert Measures


Reteachingcontinued

Remember that area is measured in square units and volume is measured in cubic units.

A covered patio measures 6.25 yards by 5 yards. What is the area of the patio in square feet?

Step 1: Find the area of the patio.

6.25 yd � 5 yd � 31.25 yd 2 So, the conversion is 31.25 yd 2 → ? ft 2 .

Step 2: Equate units.

1 yd � 3 ft So, the unit ratio is 1 yd

____ 3 ft

, or 3 ft ____ 1 yd

.

Step 3: Write the multiplication sentence. Then multiply.

31.25 yd � yd � 3 ft ____ 1 yd

� 3 ft ____ 1 yd

31.25 yd � yd � 3 ft ____ 1 yd

� 3 ft ____ 1 yd

Cancel out common factors.

� 31.25 � 3 ft � 3 ft _____________ 1 Multiply.

� 281.25 ft 2 Write the area in square feet.

The area of the outdoor patio is 281.25 square feet.

Practice

4. Mr. Greene’s yard is 50 feet by 20 feet. He wants to buy sod to cover his yard. Each piece of sod is 1-yard square. What is the area of Mr. Greene’s yard in square yards?

50 ft � 20 ft � 1000 ft 2

1000 ft � ft � 1 yd ____ 3 ft

� 1 yd ____ 3 ft

� 1000 �1 yd � 1yd _____________ � 111 yd 2 9

The area of Mr. Greene’s yard is about 111 square yards.

5. An interior room is 12 feet by 17 feet. Carpet pieces are 1-yard square. How many square yards of carpet must be purchased to cover the floor of the room?

22 2 __

3 yd 2

6. A hose with a flow rate of 15 cubic feet per hour is filling a large aquarium. What is the flow rate of the hose in cubic inches per hour?

25,920 in 3 /h

8

Name Date Class


You have simplified expressions containing only numbers and operations. Now you will evaluate expressions that contain numbers and/or variables.

These expressions are called algebraic expressions.

Evaluate the expression for n � 2 and p � 1.

6p � 3n � np

Step 1: Substitute 2 for n and 1 for p in the expression.

6 � 1 � 3 � 2 � 2 � 1

Step 2: Simplify using the order of operations.

6 � 1 � 3 � 2 � 2 � 1

� 6 � 6 � 2

� 6

Evaluate the expression for a � 1 and b � 3.

2 (b � a) 3 � 5 b 2

Step 1: Substitute 1 for a and 3 for b in the expression.

2 (3 � 1) 3 � 5 � 3 2


2 (3 � 1) 3 � 5 � 3 2

� 2 (2) 3 � 5 � 3 2

� 2 � 8 � 5 � 9

� 16 � 45

� 61

PracticeComplete the steps to evaluate each expression for the given values.

1. �8c � 4a � ac; a � 3, c � �2 2. 2 y 2 � 3 x 2 � 4y; x � �3, y � 5

� 8(�2) � 4 � 3 � 3 (�2) 2 ( 5 ) 2 � 3 (�3)

2 � 4 � 5

� 16 � 12 � 6 � 2(25)� 3(9)� 20 � 22 � 50� 27 � 20 � 57

Evaluate each expression for the given values.

3. 3b � ab � 2; a � 5, b � 1 4. 5(c � d) � 6(c � 2d); c � 4, d � 1

10 61 5. 7x � 2y � 3xy; x � 5, y � 2 6. 2st � t 2 � 4s; s � �3, t � �2

61 28 7. m � p 3 � 7p; m � 10, p � 2 8. q � r 2 � 2(4 � q); q � 6, r � �1

4 25

9. A cable company charges a $36 monthly fee and then $2.99 for each movie ordered. They use the expression 36 � 2.99m, where m is the number of movies ordered, to find the total amount to charge for each month. How much would the cable company charge for the month of June if three movies were ordered?

$44.97

ReteachingEvaluating and Comparing Algebraic Expressions 9


Reteachingcontinued

Compare the expressions when a � 2 and b � �1. Use �, �, or �.

b 2 � 4ab � 3 b 3 � �2 a 3 b

Step 1: Substitute 2 for a and �1 for b in the expressions.

(�1) 2 � 4(2)(�1) � 3 (�1) 3 � �2 (2) 3 (�1)


(�1) 2 � 4(2)(�1) � 3 (�1) 3 � 2 (2) 3 (�1)

� �2 (8) (�1)

� �16 (�1)

� 16

� 1 � 4(2)(�1) � 3(�1)

� 1 � (8)(�1) � 3

� 1 � (�8) � 3

� �4

Step 3: Compare using �, �, or �.

Since �4 � 16, b 2 � 4ab � 3 b 3 � �2 a 3 b when a = 2 and b = �1.

PracticeComplete the steps to compare the expressions when x � 7 and y � 2. Use �, �, or �.

10. 2(x � y) � 3x � 0.5y � 12 x � xy

2(x � y) � 3x � 0.5y 12x � xy

� 12 (7) � (7)(2)

� 84 � 14� 98

� 2(7 � 2) � 3(7) � 0.5(2)

� 2(9) � 3(7) � 0.5(2)

� 18 � 21 � 1 � 38

2(x � y) � 3x � 0.5y � 12 x � xy when x � 7 and y � 2.

Compare the expressions for the given values. Use <, >, or �.

11. 3 a 2 b 2 � 4b �� 2 a 2 b � 5b; a � �1, b � 3

12. �2 h � (2h � k 2 ) �� h 2 � (2h � k ); h � �3, k � 3

13. 5 (x � y) 2 � 2y �� (x � y ) 2 ; x � 5, y � 2

14. k 2 � 2j �� j 3 � 2k; j � 3, k � 7

15. Cell phone company A charges a $30 monthly fee and 15 cents per minute. They use the expression 30 � 0.15m to find the total amount to charge for each month. Cell phone company B charges a $25 monthly fee and 17 cents per minute. They use the expression 25 � 0.17m to find the total amount to charge for each month. Which cell phone company charges less for 300 minutes during a month? A

9

Name Date Class


You have solved problems by adding and subtracting pairs of numbers. Now you will solve problems by adding and subtracting three or more rational numbers.

Simplify � 2 __ 7 � 6 __ 7 � 1 __ 7 � � � 3 __ 7 � .Step 1: Write the problem as addition.

� � 2 __ 7

� 6 __ 7 � � � 1 __

7 � � 3 __

7

Step 2: Group the terms with like signs.

� � 2 __ 7

� � � 1 __ 7 � � 6 __

7 � 3 __

7

Step 3: Add.

� � 3 __ 7

� 9 __ 7 � 6 __

7

Simplify 2.14 � 0.22 � 5.25 � (�3.81).

Step 1: Write the problem as addition.

� 2.14 � (�0.22) � 5.25 � (�3.81)

Step 2: Group the terms with like signs.

� 2.14 � 5.25 � (�0.22) � (�3.81)

Step 3: Add.

� 7.39 � 4.03 � 3.36

PracticeComplete the steps to simplify each expression.

1. � � 1 __ 9 � � � � 5 __

9 � � 2 __

9 � 4 __

9 2. 7.24 � (�2.78) � 3.4 � 5.12

� � � 1 __ 9 � � 5 __

9 � 2 __

9 �

� � 4 __ 9 � � 7.24 � (�2.78) � ( �3.4 ) � 5.12

� � � 1 __ 9 � �

� � 4 __ 9 � � 5 __

9 � 2 __

9 � 7.24 � 5.12 � (�2.78) � ( �3.4 )

�

� � 5 __ 9 � � 7 __

9 � 2 __

9 � 12.36 � 6.18 � 6.18

Simplify.

3. 1 __ 5

� 4 __ 5

� � � 3 __ 5

� � 0 4. � � � 3 __ 11

� � 4 __ 11

� � 5 __ 11

� � 2 ___ 11

5. � 2 __ 7 � 5 __

7 � � � 6 __

7 � � 4 __

7 � 13 ___

7 or 1 6 __

7

6. � 3 ___ 13

� 4 ___ 13

� � � 5 __ 13

� � 3 ___ 13

� � 5 ___ 13

7. 8.43 � 5.16 � (�7.22) � 10.49 8. �22.15 � 1.56 � 29.04 � 5.33

9. �1.43 � (�2.7) � 3.14 � 1.25 � �6.02 10. 34.19 � (�21.7) � 3.79 � 15.2 � 44.48

11. Mrs. Lewis has $156 in her checking account. She writes two checks for $31.19 and $15.76 and makes one deposit for $119. What is her new balance?

$228.05

ReteachingAdding and Subtracting Real Numbers 10


Reteachingcontinued

Order the numbers from least to greatest.

7 ___ 10

, �0.6, 2 __ 5

, 1

Step 1: Place each number on a number line.

1-1 0-2

-0.6 125

710

Step 2: To order the numbers, read the numbers on the number line from left to right.

�0.6, 2 __ 5

, 7 ___ 10

, 1

PracticeComplete the steps to order the numbers from least to greatest.

12. �1.75, 0.3, � 1 __ 4 , � 4 __

5 13. 3 __

5 , �1.25, � 5 ___

10 , 0.4

1-1 0-2

-1.7514

- 0.345

-

1-1 0-2

0.4-1.25

510

-

35

The numbers in order from least to greatest The numbers in order from least to

are �1.75, � 4 __ 5 , � 1 __

4 , 0.3. greatest are �1.25, � 5 ___

10 , 0.4, 3 __

5 .

Order from least to greatest.

14. � 1 __ 3

, � 4 __ 3

, �1.4, �2 15. �0.9, 9 ___ 10

, �1, 0.05

�2, �1.4, � 4 __ 3 , � 1 __

3 �1, �0.9, 0.05, 9 ___

10

16. �5.8, �7, �3 5 __ 8 , �3.025 17. � 3 __

2 , 5 __

6 , �1.2, 0.8

�7, �5.8, �3 5 __ 8 , �3.025 � 3 __

2 , �1.2, 0.8, 5 __

6

18. �2 3 __ 4

, 7 __ 5 , 1.45, �2.7 19. �1.75, 8 ___

10 , �1.5, 0.3

�2 3 __ 4 , �2.7, 7 __

5 , 1.45 �1.75, �1.5, 0.3, 8 ___

10

20. Terra is deep-sea diving. She descends 20 feet below sea level, ascends 7 feet, descends 15 feet and ascends 4 feet. What is her current position in relation to sea level?

24 ft below sea level or �24 ft

10

Name Date Class

© Saxon. All r ights reserved. 21 Saxon Algebra 1

The probability of an event is the likelihood that the event will occur. You can estimate the probability of an event by performing an experiment. An experiment is an activity involving chance. The more trials you perform, the more accurate your estimate will be.

experimental probability � number of times an event occurs ___________________________ number of trials

An experiment consists of randomly selecting marbles from a bag. Use the results in the table to find the experimental probability of each event.

a. selecting a green marble

number of times an event occurs ___________________________ number of trials

� 8 ______________ 12 � 8 � 15 � 5

� 8 ___ 40

� 1 __ 5

b. not selecting a white marble


� 12 � 8 � 5 ______________ 12 � 8 � 15 � 5

� 25 ___ 40

� 5 __ 8

PracticeAn experiment consists of randomly selecting pens from a bag. Use the results in the table and complete the steps to find the experimental probability of each event.

1. selecting a blue pen


� 9 ______________ 15 � 5 � 9 � 7

� 9 ___ 36

� 1 __ 4

2. not selecting a red or a blue pen


� 15 � 7 ______________

15 � 5 � 9 � 7 �

22 ___

36 �

11 ___

18

An experiment consists of selecting letters from a bag. Use the results in the table to find the experimental probability of each event.

3. Selecting the letter M 8 ___ 45

4. Not selecting the letter B 7 __ 9

5. Selecting a vowel 2 ___ 15

6. Not selecting L or E 29 ___ 45

Outcome Frequency Red 12

Green 8White 15Blue 5

Outcome Frequency Black 15Red 5Blue 9

Green 7

Outcome M A R B L EFrequency 8 4 7 10 14 2

ReteachingDetermining the Probability of an Event

INV

1

© Saxon. All r ights reserved. 22 Saxon Algebra 1

Reteachingcontinued

You can use experimental probability to make predictions. A prediction is an estimate or guess about something that has not yet happened.

Kiro is baking cookies to sell at the track team bake sale. He wants to sell only the cookies that weigh at least 30 grams. Out of the first 24 cookies, 3 cookies weigh less than 30 grams.

a. What is the probability a cookie will weight less than 30 grams? Express the probability as a percent.


� 3 ___ 24 � 1 __ 8 � 12.5%

b. If Kiro bakes 10 dozen cookies, about how many of the cookies are likely to weigh less than 30 grams?

12.5% � (10 �12) � 0.125 � 120 � 15

15 cookies are likely to weigh less than 30 grams.

PracticeInspectors test 500 cars for air pollution emissions. Thirteen of them fail the test. Complete the steps to answer each question.

7. What is the probability that a car chosen at random will fail the test? Express your answer as a fraction and a percent.


� 13 ____ 500

� 2.6%

8. If 9000 cars are scheduled for the smog emissions test, about how many will likely fail?

2.6% � 9000 � 0.026 � 9000 � 234 234 cars are likely to

fail the test.

A machine assembles 600 boxes. An inspector determines that 594 of the boxes have no defects.

9. What is the probability that a box chosen at random will have no defects? Express your answer as a fraction in lowest terms and as a percent.

Fraction: 99 ____ 100

; Percent: 99%

10. The machine assembles 800 boxes. About how many will have no defects?

792 boxes

INV

1

Name Date Class


You have used multiplication and division to solve problems with whole numbers. Now you will use multiplication and division to solve problems with signed numbers.

Multiplying and Dividing Signed Numbers

The product or quotient of two numbers with the same sign is a positive number.The product or quotient of two numbers with opposite signs is a negative number.

Simplify the expression 6(�3). Justify your answer.

Step 1: Determine the sign (� or �) for the product. The product of two numbers with opposite signs is negative.

6(�3) � �

Step 2: Multiply.

6(�3) � � 18 � �18

Step 3: Justify your answer.Multiplying two numbers with opposite signs results in a negative product.

Simplify the expression �10 � (�2). Justify your answer.

Step 1: Determine the sign (� or �) for the quotient. The quotient of two numbers with the same sign is positive.

�10 � (�2) � �

Step 2: Divide.

Step 3: Justify your answer.Dividing two numbers with the same sign results in a positive quotient.

PracticeComplete the steps to simplify each expression. Justify your answer.

1. �4(�5) 2. 16 � (�2)

The product of two numbers with the The quotient of two numbers with

same sign is positive . opposite signs is negative.

�4(�5) � 20 16 � (�2) � �8

Multiplying two numbers with the same Dividing two numbers with opposite

sign results in a positive product. signs results in a negative quotient.

Simplify each expression.

3. �6(7) � �42 4. 3(�1.6) � �4.8

5. (�12)(�5) � 60 6. �21 � 7 � �3

7. �22 � (�11) � 2 8. �42 � (7) � �6

9. Mr. Young’s monthly bank withdrawals can be represented by −$810.00. How can you represent his average withdrawal if the month had 30 days?

�$27

ReteachingMultiplying and Dividing Real Numbers 11


Reteachingcontinued

Two numbers whose product is 1 are called reciprocals.Dividing by a fraction is the same as multiplying by the reciprocal of the divisor.

Evaluate the expression � 4 __ 9 � � � 1 __

2 � .

Step 1: Determine the sign (� or �) for the quotient. The quotient of two numbers with the same sign is positive.

� 4 __ 9

� � � 1 __ 2 � � �

Step 2: Divide.

� 4 __ 9

� � � 2 __ 1 � Multiply by the reciprocal of � � 1 __

2 � .

� 4 __ 9 � � � 2 __

1 � � � 8 __

9 � 8 __

9

PracticeComplete the steps to evaluate each expression.

10. � 9 ___ 16

� 3 __ 8 11. � 5 __

6 � � � 7 __

8 �

The quotient of two numbers with opposite The quotient of two numbers withsigns is negative . the same signs is positive.

� 9 ___ 16

� 3 __ 8

� � 9 ___ 16

� 8 __ 3 � � 72 ___

48 � � 3 __

2 � 5 __

6 � � � 7 __

8 � � � 5 __

6 �

� � 8 __ 7 � � 40 ___

42 � 20 ___

21

Evaluate each expression.

12. 3 __ 4 � � � 1 __

3 � � �2 1 __

4 13. � 7 ___

10 � � � 1 __

2 � � 1 2 __

5

14. � 1 __ 7

� � � 9 ___ 12

� � 4 ___ 21

15. 1 __ 4 � � � 11 ___

12 � � � 3 ___

11

16. � 2 __ 7 � � � 1 __

3 � � 6 __

7 17. � 8 __

9 � 5 ___

10 � �1 7 __

9

18. � 2 __ 9

� 1 __ 2

� � 4 __ 9 19. 4 __

7 � 7 __

8 � 32 ___

49

20. The average annual minimum temperature in Pinecreek, Minnesota is �45°F. In Northwood, Iowa, the average annual minimum temperature is half as cold. What is the average annual minimum temperature in Northwood, Iowa?

�22.5° F

11

Name Date Class


You have simplified expressions. Now you will use the properties of addition and multiplication to simplify expressions.

The six properties below are true for all real numbers.

Identity Property of Addition

Adding a and zero equals a.

a � 0 � a

12 � 0 � 12

Identity Property of Multiplication

Multiplying a by 1 equals a.

a � 1 � a

4 � 1 � 4

Commutative Property of Addition

Order does not affect the sum of a and b.

a � b � b � a

2 � 8 � 8 � 2

Commutative Property of Multiplication

Order does not affect the product of a and b.

a � b � b � a

6 � 3 � 3 � 6

Associative Property of Addition

Grouping does not affect the sum of numbers.

� a � b � � c � a � � b � c �

� 2 � 3 � � 4 � 2 � � 3 � 4 �

Associative Property of Multiplication

Grouping does not affect the product of numbers.

� a � b � � c � a � � b � c �

� 2 � 3 � � 4 � 2 � � 3 � 4 �

PracticeComplete the steps to identify each property illustrated.

1. (a � b) � c � a � (b � c) 2. a � 1 � a

(2 � 7) � 5 � 2 � (7 � 5) 8 � 1 � a

Grouping does not affect the product Multiplying by 1 equals a.

of the numbers. Identity Property of Multiplication Associative Property of Multiplication

Identify each property illustrated.

3. 4 � 9 � 9 � 4 4. (5 � 4) � 6 � 5 � (4 � 6)

Commutative Property of Associative Property ofMultiplication Addition

5. 4 � 1 � 4 6. (8 � 9) � 3 � 8 � (9 � 3)

Identity Property of Associative Property ofMultiplication Multiplication

ReteachingUsing the Properties of Real Numbers to Simplify Expressions 12


Reteachingcontinued

Simplify the expression 12x � 1 ___ 12

. Justify each step.

Step 1: 12x � 1 ___ 12

Step 2: 1 ___ 12

� 12x Commutative Property of Multiplication

Step 3: � 1 ___ 12

� 12 � � x Associative Property of Multiplication

Step 4: 1 � x � x Identity Property of Multiplication

PracticeComplete the steps to simplify the expression. Justify each step.

7. 12 � 4x � 16 8. 1 __ 3 x � 3

12 � 4x � 16 1 __ 3 x � 3

12 � 16 � 4x Commutative Property of Addition 3 � 1 __

3 x Commutative Property

28 � 4x Simplify. of Multiplication

x Simplify.

Simplify each expression. Justify each step.

9. 23 � 5x � 7 10. 1 ___ 12

x � 4

23 � 5x � 7 1 ___ 12

x � 4

23 � 7 � 5x CommutativeProp. of Add.

4 � 1 ___ 12

x Commutative Prop. of Mult.

30 � 5x Simplify. 1 __ 3 x Simplify.

11. 12 � 4x � 6 � 3x 12. 3 __ 7 x � 21

12 � 6 � 4x � 3x CommutativeProp. of Add.

21 � 3 __ 7 x Commutative

Prop. of Mult.

18 � 7x Simplify. 9x Simplify.

12

Name Date Class


You have used exponents to find the square of a number. Now you willestimate the square root of a number and compare the values of square roots.

Estimate the value ��

15 to the nearest integer.

Step 1: 15 is not a perfect square.

Step 2: 15 is between the perfect squares 9 and 16.

Step 3: ��

15 is between 3 and 4 because �

� 9 � 3 and �

� 16 � 4

Step 4: ��

15 is closer to the number 4 because 15 is closer to 16 than 9.

Estimate the value ��


Step 1: 40 is not a perfect square.

Step 2: 40 is between the perfect squares 36 and 49.

Step 3: ��

40 is between 6 and 7 because �

� 36 � 6 and �

� 49 � 7.

Step 4: ��

40 is closer to the number 6 because 40 is closer to 36 than 49.

PracticeComplete the steps to estimate the square root of each number.

1. Estimate ��

7 to the nearest integer. 2. Estimate ��


7 is not a perfect square. 79 is not a perfect square.

7 is between the perfect squares 4 and 9 . 79 is between the perfect squares 64 �

� 7 is between 2 and 3 because and 81 .

��

4 � 2 and ��

9 � 3 . ��

79 is between 8 and 9 because

��

7 is closer to the number 3 because 7 ��

64 � 8 and ��

81 � 9 .

is closer to 9 than 4 . ��

79 is closer to the number 9

because 79 is closer to 81 than 64 .Estimate the square root of each number.

3. ��

5 4. ��

8

2 3 5. �

� 34 6. �

� 105

6 10 7. �

� 84 8. �

� 77

9 9

9. ��

24 10. ��

45

5 7

11. ��

119 12. ��

228

11 15

ReteachingCalculating and Comparing Square Roots 13


Reteachingcontinued


��

9 � ��

25 � ��

16 � ��

16

Step 1: Simplify the expression.

3 � 5 � 4 � 4

Step 2: Add.

8 � 8

Step 3: Compare.

8 � 8


��

81 � ��

36 � ��

100 � ��

1

Step 1: Simplify the expression.

9 � 6 � 10 � 1

Step 2: Add.

15 � 11

Step 3: Compare.

15 � 11

PracticeComplete the steps to compare the expressions. Use �, �, or �.

13. ��

49 � ��

4 � ��

25 � ��

36 14. ��

400 � ��

121 � ��

289 � ��

196

Step 1: Simplify the expression. Step 1: Simplify the expression.

7 � 2 � 5 � 6 20 � 11 � 17 � 14

Step 2: Add. Step 2: Add.

9 � 11 31 � 31

Step 3: Compare. Step 3: Compare.

9 � 11 31 � 31


15. ��

25 � ��

64 � ��

49 � ��

9 16. ��

625 � ��

25 � ��

400 � ��

9

13 � 10 30 � 23

17. ��

49 � ��

81 � ��

144 � ��

16 18. ��

169 � ��

144 � ��

81 � ��

289

16 � 16 25 � 26

19. ��

144 � ��

64 � ��

121 � ��

81 20. ��

100 � ��

361 � ��

256 � ��

324

20 � 20 29 � 34

21. The area of a square classroom is 576 square feet. The area of the square classroom is four times the area of the square teacher’s room. What are the dimensions of the teacher’s room? 12 ft � 12 ft

13

Name Date Class


You have multiplied and divided fractions. Now you will use fractions to determine the likelihood of an event.

A bag contains 2 red marbles, 3 blue marbles, and 5 yellow marbles. What is the probability of randomly choosing a yellow marble?

Step 1: P(yellow) � 5 yellow marbles

______________ 10 marbles in all

Step 2: P(yellow) � 5 ___ 10

Step 3: 5 ___ 10

� 0.5 � 50%

A bag contains 2 red marbles, 3 blue marbles, and 5 yellow marbles. What is the probability of randomly not choosing a yellow marble?

Step 1: P(not yellow) � 2 red � 3 blue ______________ 10 marbles in all

Step 2: P(not yellow) � 5 ___ 10

Step 3: 5 ___ 10

� 0.5 � 50%

PracticeComplete the steps to find the probability. Use the information from the examples above.

1. P(red) 2. P(red or yellow)

P(red) � 2 red marbles ______________ 10 marbles in all

P(red or yellow) � 2 red � 5 yellow

______________ 10 marbles in all

P(red) � 2 ___

10 P(red or yellow) � 7 ____

10

2 ___ 10

� 0.2 � 20 % 7 ___

10 � 0.7 � 70 %

Refer to the spinner at the right.Find the probability of each event.

A

B C

D

E

FG

H

3. P(A) 1 __ 8

4. P(B) 1 __ 8

5. P(D or E) 1 __ 4

6. P(not H) 7 __ 8

7. P(A, B, or C) 3 __ 8

8. P(not G or not F) 3 __ 4

ReteachingDetermining the Theoretical Probability of an Event 14


Reteachingcontinued

You roll a number cube one time. What is the probability that you roll a number less than 3?

Step 1: 1 and 2 are the numbers less than 3 on a number cube.

Step 2: P(1 or 2) � 2 __ 6 � 1 __

3

You roll a number cube one time. Do you have a greater chance of rolling a 3 or a 5?

Step 1: P(3) � 1 __ 6

Step 2: P(5) � 1 __ 6

Step 3: 1 __ 6

� 1 __ 6 , so the chance of rolling a 3 is

the same as the chance of rolling a 5.

PracticeA bag contains 4 red marbles, 5 blue marbles, 3 white marbles, and 3 green marbles. Complete the steps to find the probability.

9. P(red or blue) 10. Do you have a greater chance of picking a red marble or a white marble?

There are 4 red marbles and 5 blue marbles. P(red) � 4 ____ 15

P(red or blue) � 9 red or blue marbles ____________________

15 marbles in all P(white) �

3 ____ 15

P(red or blue) � 9 ____ 15

� 3 ____ 5

4 ___ 15

� 3 ____ 15

, so you have greater chance

of picking a red marble than picking

a white marble.

A bag contains 4 red marbles, 5 blue marbles, 3 white marbles, and 3 green marbles.

11. P(blue or white) 8 ___ 15

12. P(not green)

12 ___ 15

� 4 __ 5

13. Do you have a greater chance of picking a red marble or a blue marble? blue

14. Do you have a greater chance of picking a white marble or a blue marble? blue

15. Do you have a greater chance of picking a green marble or a white marble? equal

14

Name Date Class


You have used the Identity, Commutative, and Associative Properties of real numbers to simplify expressions. Now you will also use the Distributive Property to simplify numeric expressions.

The Distributive Property

For all real numbers a, b, c,

a(b � c) � ab � ac and a(b � c) � ab � ac

Simplify 2(5 � 7).

2(5 � 7)

� 2(5) � 2(7) Distribute the 2.

� 10 � 14 Multiply.

� 24 Add.

Simplify 3(6 � 4).

3(6 � 4)

� 3(6) � 3(4) Distribute the 3.


� 6 Subtract.

Simplify. �(2 � 6).Rewrite the expression as �1(2 � 6), then distribute.

�1(2 � 6)

� (�1)(2) � (�1)(6) Distribute the �1.

� �2 � 6 Multiply.

� �8 Subtract.

Simplify. �4(�3 � 7).

�4(�3 � 7)

� (�4)(�3) � (�4)(�7) Distribute the �4.


� 40 Add.


1. 5(3 � 4) 2. �6(7 � 3)

5(3 � 4) �6(7 � 3)

� 5 (3) � 5 (4) Distribute the 5 . � � �6 � (7) � � �6 � (3) Distribute the �6

� 15 � 20 Multiply. � �42 � 18 Multiply.

� 35 Add. � �60 Subtract.


3. 4(8 � 3) � 20 4. �3(�7 � 2) � 27

5. �3(9 � 5) � �12 6. 12(3 � 2) � 60

7. 10(�6 � 2) � �80 8. �15(5 � 2) � �45

15ReteachingUsing the Distributive Property to Simplify Expressions


Reteachingcontinued

The Distributive Property can also be used to simplify algebraic expressions.

When multiplying algebraic expressions, remember to add the exponents of powers with the same base.

Simplify �3(z � 8).

�3(z � 8)

� (�3)(z) � (�3)(8) �3 Distribute the �3.

� 3z � 24 Multiply.

Simplify (p � 7)3.

(p � 7) 3

� 3(p) � 3(7) Distribute the �3.

� 3(p) � 21 Multiply.

Simplify 2t(st � t � 4s). Multiply 2t by each of the terms in the parentheses.

2t(st � t � 4s)

� (2t)(st) � (2t)(t ) � (2t)(4s)

� 2st2 � 2t2 � 8st

Simplify �ab(ac2 � b2).Multiply �ab by each of the terms in the parentheses.

�ab(ac2 � b2)

� (�ab)(ac2) � (�ab)(�b2)

� �a2bc2 � ab3


9. (9 � y)8 10. �3p2q(pq � 2q2)(9 � y)8 �3p2q(pq � 2q2)

� 8 (9) � 8 (y) Distribute the 8 . Multiply �3p2q by each of the terms in

� 72 � 8y Multiply. the parentheses.

� � �3p2q � (pq) � � �3p2q � (�2q2)

� �3p3q2 � 6p2q3


11. (y � 15)5 � 5y � 75 12. �9(�x � 2) � 9x � 18

13. 7b(b2 � 4ac) � 7b3 � 28abc 14. �8z(2z � 6) � �16z 2 � 48z

15. (3p � pq3)2pq � 6p2q � 2p2q4 16. �5cw2(3c2 � cw2) � �15c3w2 � 5c2w4

17. Last spring members of the drama club sold tickets to the school play. They sold 300 tickets ahead of time and 250 tickets at the door. One ticket cost $6. Write an expression to show the total amount of money collected. Simplify the expression using the Distributive Property.

6(300 � 250); $3300

15

Name Date Class


You have simplified expressions that contain variables. Now you will evaluate expressions that contain variables.

To evaluate an expression that contains variables

Substitute each variable in the expression with a given numeric value.

Find the value of the expression.

Evaluate the expression for the given values of the variables.

m[�n(m � n)] for m � 5 and n � 2 5 2

m[�n(m � n)]

m[�n(m � n)]� 5[�2(5 � 2)]� 5[�2(7)]� 5[�14]� �70

PracticeComplete the steps to evaluate the expression for the given values of the variables.

1. p(5qr)

_____ �pq for p � �2, q � 3 and r � 4

p(5qr)

_____ �pq

� �2(5)(3)(4) __________

�(�2)(3)

� �120

______ 6

� �20

Evaluate each expression for the given values of the variables.

2. 5c[z(c � 2z)] for c � �2 and z � �1 �40

3. 5j(2k � j ) for j � 1 __ 2 and k � 3 __

4 2.5

4. w [x � z(w + z) � w] for x � 2, w � �1, and z � 3 3

5. (3d )(def )(2f ) for d � �3, e � 2, and f � 1 __ 3 12

Substitute each variable in the given value.Add.Multiply inside the brackets.Multiply.

Substitute each variable in the given value.

Multiply.

Divide.

ReteachingSimplifying and Evaluating Variable Expressions 16


Reteachingcontinued

A variable expression can often be simplified before it is evaluated.

First simplify s(2 � t ) � s. Then evaluate it for s � �4 and t � 2. Justify each step.s(2 � t ) � s

� 2s � st � s

� 3s � st

� 3(�4) � (�4)(2)

� �12 � 8

� �4

First simplify x(2y � x) � 3xy. Then evaluate it for x � �5 and y � 3. Justify each step.

x(2y � x) � 3xy

� 2xy � x 2 � 3xy

� �xy � x 2

� �(�5)(3) � (�5) 2

� �(�5)(3) � 25

� 15 � 25

� 40

PracticeComplete the steps to simplify each expression. Then evaluate for p � �3 and q � 2. Justify each step.

6. pq � 3q(3 � 2p) � q 7. 2q(p � q) � 3pq

� pq � 9q � 6pq � q � 2pq � 2 q 2 � 3pq

� �5pq � 8q � 5pq � 2 q 2

� �5(�3 ) (2 ) � 8(2 ) � 5( �3 ) (2 ) � 2(2 ) 2

� 30 � 16

� 46 � 5(�3 ) (2 ) � 2(4 )

� �30 � 8

� �22

Simplify the expression. Then evaluate for x � 4 and y � �2. Justify each step

8. 4x(3 � y) � 2x � xy

Sample: 4x (3 � y) � 2x � xy

� 12x � 4xy � 2x � xy

� 10x � 3xy

� 10(4) � 3(4)(�2)

� 40 � 24

� 16

Distributive Property

Combine like terms.

Substitute.

Multiply.

Add.

Distributive Property

Combine like terms.

Substitute.


Multiply.

Add.

Distribute the 3q

Combine

Like terms .

Substitute

Multiply .

Add .

Distribute the 2qCombine

Like terms .

Substitute.

Evaluate.

the exponent .

Multiply .

Add .

Distribute the 4x.

Combine like terms.

Substitute.

Multiply.

Subtract.

16

Name Date Class


You have simplified and evaluated variable expressions. Now you will translate words into algebraic expressions.An algebraic expression, or variable expression, is an expression that contains at least one variable.

Translating Words into OperationsWords Operation

sum, total, more than, added, increased by, plus Additionless, minus, decreased by, difference, less than Subtractionproduct, times, multiplied Multiplicationquotient, divided by, divided into Division

Write an algebraic expression for the phrase x plus 3.

Find the word “increased by” in the table. It means addition. x � 3

Write an algebraic expression for the phrasethe quotient of r and 11.

Find the word “quotient” in the table. It means division. r � 11

PracticeComplete the steps to write an algebraic expression for each phrase.

1. the product of 7 and t 2. z decreased by 9

Find the word “product” in the table. Find the words “decreased by” in

It means multiplication. the table. They mean subtraction .

7 � t z � 9

Write an algebraic expression for each phrase.

3. the total of m and 10 4. 16 divided by w

m � 10 16 � w

5. the difference of q and 12 6. 17 multiplied by s

q � 12 17 � s

7. Krista owns 3 more books than Laurie, who owns b books. Write the expression that shows the number of books Krista owns.

b � 3

ReteachingTranslating Between Words and Algebraic Expressions 17


Reteachingcontinued

You can also translate algebraic expressions into words. Use the table on the previous page to find the operation represented by the symbol. Then replace the symbol with the appropriate word(s).

Use words to write the algebraic expression r � 5 in two different ways.

Find the subtraction operation in the table.

Replace the subtraction symbol with the appropriate word(s).

r minus 5

r decreased by 5

Use words to write the algebraic expression 9 � d in two different ways.

Find the multiplication operation in the table.

Replace the multiplication symbol with the appropriate word(s).

the product of 9 and d

9 times d

PracticeComplete the steps to use words to write each algebraic expression in two different ways.

8. x � 15 9. 21 � z

Find the addition operation in the table. Find the division operation in the table.

Replace the addition symbol. Replace the division symbol.

x plus 15 21 divided by z

x increased by 15 the quotient of 21 and z

Use words to write each algebraic expression in two different ways.

10. 21 ___ k 11. 2v � 1

Sample: the quotient of Sample: 1 more than 2 times

21 and k, 21 divided by k v, the sum of 2 times v and 1

12. 2v � 1 13. 1 __ 3 w � 8

Sample: 1 more than

Sample: one-third w 2 times v, the sum of minus 8, the difference

2 times v and 1 of one-third w and 8

Write a sentence that could be described by each expression.

14. 2p � 3 15. 1 __ 3 a � 4

Sample: Erin has 3 more Sample: Ryan’s sister

than twice as many borrowed one-third of his

paperclips as Janet. allowance plus $4.

17

Name Date Class


You have learned that algebraic expressions are made up of terms. Now you will learn about like and unlike terms in algebraic expressions.

Like terms are two or more terms that have the same variable or variables raised to the same power. Unlike terms are two or more terms with different variables, or with the same variable or variables raised to a different power.

Example Type

3x, 8x, 99x Like Terms

5pqr, 24pqr, 955pqr Like Terms

2bc, 2cd, 2de Unlike Terms

3fg, 3fh, 3fj Unlike Terms

Simplify the expression 16a � 9a.

16a � 9a

� (16 � 9)a Distribution Property

� 7a Simplify.

Simplify the expression 7z � (�2z) � (3z).

7z � (�2z) � (3z)

� (7 � 2 � 3)z Distribution Property

� 2z Simplify.


1. �7d � 3d � (�2d) 2. 3pq � 4p � 2qp

�7d � 3d � (�2d) 3pq � 4p � 2qp

� (�7 � 3 � 2)d Distributive Property � 3pq � 2qp � 4p Rearrange the terms.

� �2d Simplify. � 3pq � 2pq � 4p Rearrange the factors.

� (3 � 2)pq � 4p Distributive Property

� 5pq � 4p Simplify.


3. �9a � (�3a) � 2a 4. pq � 3pq � 8pq

�4a �10pq

5. �3gh � 2fgh � 11gh 6. 5wx � 7k � 2xw

8gh � 2fgh 7wx � 7k

ReteachingCombining Like Terms 18


Reteachingcontinued

You can also combine like terms with exponents.

Example Type

5 xy 2 , 12 xy 2 , 44x y 2 Like Terms

5 y 4 , 24 y 4 , 955 y 4 Like Terms

2x, 2 x 2 , 2 x 3 Unlike Terms

3 a 2 , 3 b 2 , 3 c 2 Unlike Terms

Simplify the expression 2 m 3 n � n 2 � n m 3 � 5 n 2 .

2 m 3 n � n 2 � n m 3 � 5 n 2

� 2 m 3 n � nm 3 � n 2 � 5 n 2 Rearrange the terms.

� 2 m 3 n � m 3 n � n 2 � 5 n 2 Rearrange the factors.

� (2 � 1) m 3 n � (1 � 5) n 2 Use the Distributive Property.

� 3 m 3 n � 6 n 2 Simplify.

PracticeComplete the steps to simplify the expression.

7. � a 3 b 2 � 4ba � 3 b 2 a 3 � 5ab

� a 3 b 2 � 4ba � 3 b 2 a 3 � 5ab

� � a 3 b 2 � 3 b 2 a 3 � 4ba � 5ab Rearrange the terms.

� � a 3 b 2 � 3 a 3 b 2 � 4ab � 5ab Rearrange the factors.

� (�1 � 3) a 3 b 2 � (4 � 5) ab Use the Distributive Property.

� 2 a 3 b 2 � ab Simplify.


8. jk � jk 2 � 4jk � 2j k 2 9. �6 c 2 d � 4cd � 12 dc 2

�3jk � 3j k 2 6 c 2 d � 4cd

10. 2 x 4 � 3 x 2 � 7 x 4 � qx 2 � x 4 11. �ab � ab 2 � 4ba � 16 b 2 a

�4 x 4 � 3 x 2 � q x 2 �5ab � 15a b 2

12. Emily is installing a rectangular vegetable garden in her yard. She wants the length of the garden to be 5 feet longer than 2 times the width. The diagram represents the measurements of her garden. Find the perimeter of the garden as a simplified variable expression. Then evaluate the expression for w � 15 feet.

2w + 5

w

P � 6x � 10; P � 100 ft

18

Name Date Class


You have used addition and subtraction to combine real numbers. Now you will use addition and subtraction to solve one-step equations.

An equation is a statement containing an equal sign. The two quantities on either side of the equal sign are equal. A solution of an equation that contains one variable is a value of the variable that makes the equation true.

After you have solved an equation, substitute the solution for the variable to make sure you are correct.

State whether the value of the variable is a solution of the equation.

y � 7 � 12 for y � 5

y � 7 � 12

(5) � 7 � 12 Substitute 5 for y.

12 � 12 ✓


c � 4 � 11 for c � 7

c � 4 � 11

(7) � 4 � 11 Substitute 7 for c.

3 � 11 ✕

PracticeComplete the steps to determine if the value of the variable is a solution of the equation.

1. 14 � z � 8 for z � �6 2. 9 � m � 13 for m � 4

14 � z � 8 9 � m � 13

14 � (�6) � 8 Substitute �6 for z. 9 � m � 13

14 � 6 � 8 9 � ( 4 ) � 13 Substitute 4 for m.

20 � 8 13 � 13


3. a � 5 � 4 for a � 9 4. 10 � q � 3 for q � 6

solution, 9 � 5 � 4 not a solution, 10 �6 � 3

5. 15 � 20 � h for h � �5 6. 4 � x � �2 for x � �6

not a solution, 15 � 20 � 5 solution, 4 � 6 � �2

7. t � 2 � �4 for t � �2 8. 5 � p � �5, for p � 10

not a solution, �2 � 2 � �4 solution, 5 � 10 � �5

9. g � 3 � �5 for g � �2 10. 8 � w � 3 for w � 12

solution, �2 � 3 � �5 not a solution, 8 � 12 � 3

ReteachingSolving One-Step Equations by Adding or Subtracting 19


Reteachingcontinued

You can use properties of equality and inverse operations to find the solution of an equation.

Properties of Equality

Addition Property of Equality

If you add the same number to both sides of an equation, the equation will still be true.

Subtraction Property of Equality

If you subtract the same number from both sides of an equation, the equation will still be true.

Inverse operations are operations that undo each other. To solve an equation, use inverse operations to isolate the variable on one side of the equal sign. You must use the same operation on each side.

Solve w � 2 � 9. Then check the solution.

w � 2 � 9

_ � 2 � _ � 2 Add 2 to both sides to undo

w � 11 the subtraction.

Check Substitute 11 for w.

w � 2 � 9

(11) � 2 � 9

9 � 9 ✓

Solve f � 4 � 12. Then check the solution.

f � 4 � 12

_ �4 � _ �4 Subtract 4 from both sides to

f � 8 undo the addition.

Check Substitute 8 for f.

f � 4 � 12

(8) � 4 � 12

12 � 12 ✓

PracticeComplete the steps to solve the equation. Then check the solution.

11. z � 8 � 6 Check Substitute �2 for z.

z � 8 � 6 z � 8 � 6

�8 � �8 Subtract 8 from both sides (�2) � 8 � 6

z � �2 to undo the addition. 6 � 6 ✓

Solve. Then check the solution.

12. v � 11 � 3 13. d � 13 � 5

v � 14; 14 � 11 � 3 d � �8;�8 � 13 � 5

14. �12 � q � 7 15. a � 1 __ 3

� 1 __ 6

q � �5;�12 � �5 � 7 a � 1 __ 2 ; 1 __

2 � 1 __

3 � 1 __

6

16. A video game is on sale for $45, which is $14 off its regular price. What is the regular price of the video game? $59

Inverse Operations

Add Subtract

Multiply Divide

Inverse Operations

Add Subtract

Multiply Divide

19

Name Date Class


You have used number lines to plot real numbers. Now you will graph real numbers on a coordinate plane.

An ordered pair, or two numbers in parentheses, identifies a point on the plane. It is written as (x, y).

The first number is the x-coordinate. It is the distance of the point left or right of the origin. If negative, it is to the left of the origin.

The second number is the y-coordinate. It is the distance of the point above or below the origin. If negative, it is below the origin.Graph (2, 3) on a coordinate plane. Label the point.

Start at the origin. Move 2 to the right. Move 3 up.

Graph (�4, 2) on a coordinate plane. Label the point.

Start at the origin. Move 4 to the left. Move 2 up.

PracticeComplete the steps to graph each ordered pair on a coordinate plane.

1. (2, 4). 2. (�3, �2)

Start at the origin. x

y

4

-4

-4

4

O

(2,4) Start at the origin. x

y

4

-4

-4

4

O

(-3,-2) Move 2 to the right. Move 3 to the left.

Move 4 up. Move 2 down

Graph each ordered pair on a coordinate plane. Label each point.

3. (�3, 4) 4. (�3, �5)

O x

y6

-6 -4 4 6

4

-4

-6

(-3,4)

O x

y6

-6 -4 4 6

4

-4

-6

(-3,-5)

5. (0, �2) 6. (3, 0)

O x

y6

-6 -4 4 6

4

-4

-6

(0,-2)

O x

y6

-6 -4 4 6

4

-4

-6

(3,0)

x

y4

2

2 4

-2

-2-4

-4

O

IVQuadrant

IIIQuadrant

IIQuadrant

IQuadrant

(0,0)

x

y

4

-4

-4

4

O

(2,3)

x

y

4

-4

-4

4

O

(-4,2)

ReteachingGraphing on a Coordinate Plane 20


Reteachingcontinued

When there is a relationship between two variable quantities, the dependent variable always depends on the value chosen for the independent variable.

Variables

Independent variable: The variable whose value can be chosen. (Input variable)

Dependent variable: The variable whose value is determined by the input value of another variable. (Output variable)

Complete the table for the equation y � 2x � 1. Since the values of x are given, x is the independent variable. The dependent variable is y.Substitute �3 for x in the equation.

y � 2x � 1 Write the equation.

y � 2(�3) � 1 Substitute �3 for x.

y � �6 � 1 Evalute.

y � �7Substitute the other values to complete the table.

PracticeComplete the steps to fill in the table for the equation.

7. Complete the table for the equation y � �3x � 6

Substitute �1 for x in the equation.

y � �3x � 6 Write the equation.

y � �3(�1) � 6 Substitute �1 for x.

y � 3 � 6 Evalute.

y � 9 Substitute the other values to complete the table.

Complete the table for each equation.

8. y � 5x � 2 9. y � �4x � 9

10. Anita knits scarves. The equation y � 20x � 80 represents her earnings, where x is the number of scarves sold and y is the money earned. Find the amount of money Anita earns when 5, 10, 15, and 20 scarves are sold. Make a graph to represent the equation y � 20x � 80.

20, 120, 220, 320

x y

�3 �7

�1 �3

0 �1

1 __ 2 0

x y

�1 90 61 32 0

x �2 �1 1 __ 5 1

y �8 �3 3 7

x �1 0 3 4

y 13 9 �3 �7

20

y

xO

200

300

400

100

5 10 15 20

reteaching 6 subtracting real · pdf filethen follow the rules for adding real numbers. find...

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