Algebra 1 2016-17 SUMMER PACKET Ms. Bank · Algebra 1 2016-17 SUMMER PACKET Ms. Bank ... If...

[email protected] Algebra 1 Summer Packet

Algebra 1 2016-17 SUMMER PACKET Ms. Bank

Just so you know what to expect next year ……

We will use the same text that was used this past year: Algebra 1 published by McDougall Littell ISBN-13:978-0-6185-9402-3.

Summer Packet

To help students retain the math skills listed on the following page, the math department requires rising Algebra 1 students to complete this Algebra Summer Math Packet. The skills required to answer the questions in this packet are ones that should have been mastered by students in previous math courses. They are the skills covered in the first two chapters of the Algebra textbook. The packet contains a brief review and example problems for each skill.

Students are to complete all of the questions in the Practice and Quiz sections for each skill.

Please note: 1. Working through these problem sets is mandatory.

2. Students are to bring completed problem sets with them to turn in on the first day theirmath class meets next school year.

3. All work is to be done in pencil on notebook paper. ALL WORK MUST BE SHOWN.

4. Students should NOT use calculators on any portion of the packet.

5. Students should check their work upon completion of each section. Answers to thequestions are located in the back of the packet. If an answer is incorrect, studentsshould return to the work shown, attempt to find the source of the error(s), and correctthe problem.

6. If students need more explanation and practice, they should read the correspondingsection in the first or second chapter of the text and work odd numbered problems tocheck for understanding. (Answers to odd numbered problems are in the back of thetext.)

7. Students will be given a homework grade for completion of this packet.

8. Students will take a test covering this material in the first week of school. Remember itis truly beneficial to keep your mathematical mind oiled over the summer.

Should you lose this packet, it can be found on my school web page. Should you have questions, I will be checking email throughout the summer.

WORK IS TO BE DONE ON NOTEBOOK PAPER IN PENCIL. You will not turn in your copy of the packet, just your work.

[email protected] Algebra 1 Summer Packet

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Algebra 1Benchmark 1 Chapters 1 and 2 1

BENCHMARK 1(Chapters 1 and 2)

A. Expressions, Equations, and Inequalities (pp. 1–3)

A variable is a letter used to represent one or more numbers. An algebraic expressionis made from numbers, variables, and algebraic operations. The following examples describe how expressions can be evaluated, combined, written, and used to write algebraic equations and inequalities.

1. Evaluate ExpressionsEvaluate an expression Substitute a number for the variable, perform the operation(s), and simplify the result if necessary.

Evaluate the expression.

a. 16n when n 5 4 b. 25}k

when k 5 5 c. h 2 8 when h 5 12.2

d. 4}3 1 h when h 5

1}3 e. x3 when x 5 4 f. a2 when a 5 1.2

Solution:

a. 16n 5 16 4 b. 25}k

5 25}5 c. h 2 8 5 12.2 2 8

5 64 5 5 5 4.2

d. 4}3 1 h 5

4}3 1

1}3 e. x3 5 43 f. a2 5 1.22

5 5}3 5 4 4 4 5 (1.2)(1.2)

5 64 5 1.44


1. 5b when b 5 6 2. 42}h

when h 5 14

3. 14 2 b when b 5 11.3 4. v 1 7}6 when v 5

1}3

5. y4 when y 5 3 6. q2 when q 5 2.1

2. Order of OperationsOrder of operations Established rule for evaluating an expression involving more than one operation:

Step 1: Evaluate expressions inside grouping symbols.

Step 2: Evaluate powers.

Step 3: Multiply and divide from left to right.

Step 4: Add and subtract from left to right.

EXAMPLE

PRACTICE

When a number is followed directly by a variable, the operation ofmultiplication is always implied.

Vocabulary

Vocabulary

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Algebra 1Benchmark 1 Chapters 1 and 22


a. 3 24 2 5 6 b. 4(32 1 5) c. 5[12 2 (4 1 5)]

Solution:

a. 3 24 2 5 6 5 3 16 2 5 6 Evaluate power.

5 48 2 30 Multiply.

5 18 Subtract.

b. 4(32 1 5) 5 4(9 1 5) Evaluate power.

5 4(14) Add within parentheses.

5 56 Multiply.

c. 5[12 2 (4 1 5)] 5 5(12 2 9) Add within parentheses.

5 5(3) Subtract within brackets.

5 15 Multiply.


7. 4(10 2 3) 2 5 2 8. 21 1 (32 2 4) 9. 2[42 4 (9 2 3)]

3. Write ExpressionsTranslate verbal phrases into expressions.

a. The product of 8 and m increased by 5

b. The quotient of 8 and the difference of a number x and 2

c. The sum of 20 and the square of a number n

Solution:

a. 8m 1 5 b. 8

}x 2 2

c. 20 1 n2

Translate the verbal phrases into expressions.

10. The quotient when the quantity of a number y increased by 4 is divided by 6

11. 4 less than twice the square of a number q

12. 8 more than the product of a number w and 6

4. Write Equations and InequalitiesOpen sentence A mathematical statement that contains two expressions and a symbol that compares them.

Equation An open sentence that contains the symbol 5.

Inequality An open sentence that contains one of the symbols , , , or .

EXAMPLE

Keep a glossary of terms that describe each of the four basic operations.

PRACTICE

The multiplicationthat could be written in two steps (3 16 evaluated fi rst, followed by 5 6)is combined as one step.

PRACTICE


Vocabulary

EXAMPLE

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Write an equation or an inequality.

a. The difference of a number p and 12 is at most 15.

b. The product of 5 and a number m is 14.

c. A number x is at least 6 and less than 9.

Solution:

a. p 2 12 15 b. 5m 5 14 c. 6 x 9

Write an equation or inequality.

13. The quotient of 12 and a number q is at most 5.

14. The sum of twice a number h and 5 is the same as 23.

15. The difference of a number w and 4 is greater than 12 and no more than 20.

QuizEvaluate the expression.

1. h}3 1

1}3 when h 5 5 2.

64}b2 when b 5 4 3. 12 2

5a}4

when a 5 4


4. (42 2 3) (2 1 3) 1 1 5. 4[(22 2 3) 1 1] 6. [54 4 (6 2 3)2]2

}}8 2 2

Translate the verbal phrases into expressions.

7. The product of twice the number y and 4 increased by 8

8. The difference of 6 times the square of a number x and 15

Write an equation or inequality.

9. The sum of the number b and 12 is twice the number b.

10. The product of a number q and 3 is no less than 10 and no more than 15.


EXAMPLE

PRACTICE

“No less than” ( )and “no greater than” ( ) are opposites of “less than” ( ) and “greater than” ( ),respectively.

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B. Problem Solving (pp. 4–5)

One way to try solving a math problem is to use an organized strategy, or problem-solving plan. Read the problem to fi nd what information is given and what you need to fi nd out. Decide on the strategy you will use, and apply it to solve the problem. Finally, check that your solution makes sense.

1. Check Possible SolutionsSolution of an equation or inequality A number that can be substituted for the variable in an equation or inequality to make a true statement.

Check whether the given number is a solution of the equation or inequality.

a. 2x 2 8 5 22; 3 b. x}3 1 1 5 7; 6 c. x 2 5 3; 2

Solution:

a. 2(3) 2 8 0 22 b. 6}3 1 1 0 7 c. 2 2 5 3

6 2 8 0 22 2 1 1 0 7 23 3

22 5 22 3 Þ 7

3 is a solution. 6 is not a solution. 2 is a solution.

Check whether the given number is a solution of the equation or inequality.

1. 5 1 a 10; 24 2. n 2 3}

12 5 1; 4 3.

r}4 1 3 5 5; 22

4. 28p 2 6 0; 21 5. 9d 2 3 5 60; 7 6. m 1 8 27; 214

2. Read and Understand a ProblemRead the problem below. Identify what you know and what you need to fi nd out. You do not need to solve the problem.

You run in a city where the short blocks on north-south 0.2 mi

0.03 mistreets are 0.03 miles long. The long blocks on east-west streets are 0.2 mile long. You will run 2 long blocks east, a number of short blocks south, 2 long blocks west, then back to your starting point. You want to run 1.1 miles. How many short blocks should you run?

Solution:

What do you know? Each short block is 0.03 miles long. Each long block is 0.2 miles long. You will run 4 long blocks total (2 east 1 2 west). You will run s short blocks total (south and north). You want to run a total of 2 miles.

What do you want to fi nd out? How many short blocks should you run so that the distance you run on short blocks and the distance you run on 4 long blocks makes a total of 1.1 miles?

Vocabulary

EXAMPLE

PRACTICE

EXAMPLE

There may be more than one method that can be used to solve a problem.

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7. Read the problem below. Identify what you know and what you need to fi nd out. You do not need to solve the problem.

A bicycle park has a long trail and a short trail. The long

start/finish

2 km

5 km

trail is 5 km long. The short trail is 2 km long. You will ride 3 laps on the short trail and some number of laps on the long trail. You want to ride 21 km. How many laps should you ride on the long trail?

3. Make a PlanWrite a verbal model of the statement below.

How many short blocks should you run so that the distance you run on short blocks and the distance you run on 4 long blocks makes a total of 1.1 miles?

Solution:

Distance run onshort blocks

1Distance run on

long blocks5

Totaldistance

(miles) (miles) (miles)

Length of a short block

Number of

short blocks1

Length of a long block

Number of long blocks

5Total

distance

(miles/block) (block) (miles/block) (block) (miles)

8. Write a verbal model for the problem in Exercise 7.

QuizCheck whether the given number is a solution of the equation or inequality.

1. 6 1 j 4; 21 2. n 1 5}

2 5 6; 7 3.

m}3

2 8 5 25; 9

4. 22y 1 3 0; 2 5. 4g 2 5 5 35; 10 6. b 1 12 0; 213

Read the problem below. Identify what you know and what you need to fi nd out. Then, write a verbal model of the problem. You do not need to solve the problem.

7. Jim’s grandmother exercises by walking the main

90 yd

start

20 yd

rectangular hall of a local shopping mall. She walks 90 yards down the length of the hall, turns right, and walks 20 yards across the width of the hall. Then, she turns right and walks up the length of the hall again. Finally, she turns right one more time, and walks 20 yards across the width of the hall and ends up at her starting point. Jim’s grandmother wants to walk 970 yards. She will walk the length of the hall 9 times. How many times will she walk across the width of the hall?

PRACTICE

EXAMPLE

PRACTICE


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C. Representations of Functions (pp. 6–9)

Functions can be represented by mapping diagrams, tables, verbal or algebraic function rules, and graphs. Each input value and its corresponding output value make up an ordered pair. An ordered pair can be written as (input, output) or plotted as a point on a coordinate grid.

1. Identify the Domain and Range of a FunctionFunction A pairing of input values to output values, where the value of each output depends on the value of the corresponding input, and each input corresponds to exactly one output.

Domain The set of input values for a function.

Range The set of output values for a function.

Identify the domain and range of the function.

a. Input Output

0 10

1 11

2 12

3 13

b.3 21

6 22

9 23

12 24

c.Input Output

28 24

4 2

6 3

10 5

Solution:

a. Domain: 0, 1, 2, 3 b. Domain: 3, 6, 9, 12 c. Domain: 28, 4, 6, 10

Range: 10, 11, 12, 13 Range: 21, 22, 23, 24 Range: 24, 2, 3, 5

Identify the domain and range of each function.

1. Input Output

0 1

1 3

2 5

3 7

2. Input Output

25 1

210 2

215 3

220 4

3.26 600

24 400

21 100

0 0

EXAMPLE

PRACTICE

Vocabulary

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Tell whether the pairing is a function.

a.Input Output

26 2

27 3

28 4

29 5

b.24 12

23 13

0 16

19

c.Input Output

35 0

40 5

40 10

45 15

Solution:

a. Yes b. No; 0 maps to c. No; 40 maps to two two outputs. outputs.

Tell whether the pairing is a function.

4. Input Output

214 4

28 4

22 4

4 4

5. Input Output

9 0

15 6

27 18

35 26

6.21 0

0 21

1 3

2

2. Write a Function RuleIndependent variable A function’s input variable.

Dependent variable A function’s output variable.

Write a rule for the function.

Input, x 0 1 2 3 4

Output, y 0 3 6 9 12

Solution:

Each value of y is 3 times the corresponding x value. The function rule is y 5 3x.


7. Input, x 24 22 0 2 4

Output, y 22 21 0 1 2

8. Input, x 5 10 15 20 25

Output, y 1 6 11 16 21

9. Input, x 3 5 9 12 16

Output, y 12 14 18 21 25

10. Input, x 26 22 3 7 8

Output, y 6 2 23 27 28

A function rule usually states the dependent variable y as a function of the independent variable x, such as y 5 x 1 3.

PRACTICE

EXAMPLE

Vocabulary

EXAMPLE

PRACTICE

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3. Make a Table for a FunctionMake a table for the function and identify the range of the function.

y 5 x 1 2.6

Domain: 2, 3, 4, 5, 6

Solution:

Input, x 2 3 4 5 6

Output, y 4.6 5.6 6.6 7.6 8.6

Range: 4.6, 5.6, 6.6, 7.6, 8.6

Make a table for the function and identify the range of the function.

11. y 5 2}3 x 12. y 5 x 2 1.1 13. y 5 22x 1 5

Domain: 3, 6, 9, 12, 15 Domain: 25, 24, 23, 22, 21 Domain: 1, 2, 4, 7, 9

14. y 5 x 1 1}

2 15. y 5 x 1 14 16. y 5 25x


4. Graph a FunctionGraph the function y 5 x 2 2 with domain 4, 5, 6, 7, and 8.

Solution:

Step 1: Make an input-output table.

Input, x 4 5 6 7 8

Output, y 2 3 4 5 6

Step 2: List the ordered pairs (x, y).

(4, 2), (5, 3), (6, 4), (7, 5), (8, 6)

Step 3: Plot a point for each ordered pair (x, y).

1

2

3

4

5

6

7

8y

1 x2 3 4 5 6 7 8

Graph the function.

17. y 5 x 2 3 18. y 5 1}3 x 19. y 5 3x 2 3


PRACTICE

EXAMPLE

PRACTICE

EXAMPLE

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20. y 5 1.5x 1 2 21. y 5 1}2 x 2 2 22. y 5

x 1 2}

3


QuizIdentify the domain and range of each function.

1. Input Output

7 23

10 35

14 51

17 63

2. Input Output

210 90

28 54

25 15

24 6

3.23 2

3}8

22 2 1}4

21 2 1}8

0 0

Tell whether each pairing is a function.

4. Input Output

5 8

8 5

8 4

10 3

5.16 4

24

36 26

6

6. Input Output

25 22

2 5

7 10

11 14


7. Input, x 25 23 0 4 6

Output, y 225 215 0 20 30

8. Input, x 2 4 5 7 10

Output, y 24 28 210 214 220

9. Input, x 3 6 8 10 13

Output, y 24 21 1 3 6

10. Input, x 10 24 32 48 50

Output, y 15 36 48 72 75

Make a table for the function and identify the range of the function.

11. y 5 24x 1 10 12. y 5 3}4 x 2 1 13. y 5

2x 1 3}

4


Graph the function.

14. y 5 x 2 6 15. y 5 5x 16. y 5 2x 1 4


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D. Operations (pp. 10–14)

Whole numbers, integers, and rational numbers are part of the set of real numbers. The following examples describe different operations with real numbers.

1. Find Opposites of Real NumbersOpposite of a real number a 2a (read “the opposite of a”) is the same distance from 0 on a number line as a, but it is on the opposite side of 0.

For the given value of a, fi nd 2a.

a. a 5 3 b. a 5 7 3}5 c. a 5 25.4

Solution:

a. 2a 5 2(3) b. 2a 5 217 3}5 2 c. 2a 5 2(25.4)

2a 5 23 2a 5 27 3}5 2a 5 5.4

For the given value of the variable, fi nd the opposite.

1. x 5 26.2 2. u 5 809 3. m 5 0.25

4. w 5 45}8

5. k 5 2 6

}11 6. c 5 28

1}7

2. Find Absolute Values of Real NumbersAbsolute value of a real number a ZaZ (read “the absolute value of a”) is the distance between a and 0 on a number line. If a is greater than or equal to 0, ZaZ is a. If a is less than zero, ZaZ is the opposite of a.

For the given value of a, fi nd ZaZ.

a. a 5 8 b. a 5 2 4}9 c. a 5 11.5

Solution:

a. ZaZ 5 Z8Z 5 8 b. ZaZ 5 )2 4}9 ) 5 212

4}9 2 5

4}9

c. ZaZ 5 Z11.5Z 5 11.5

For the given value of the variable, fi nd the absolute value.

7. b 5 0.4 8. y 5 250 9. p 5 21.6

10. v 5 29 11. n 5 23}5

12. h 5 10 8}9

3. Add Real NumbersSum The result of adding two or more real numbers.

To use a number line to fi nd the sum of a 1 b:

• Start at a.

• If b 0, you will move to the right. If b 0, you will move to the left.

• Find ZbZ and move that many units.

• The number you stop on is the sum.

EXAMPLE

Note that 2a is postive when a is negative.

PRACTICE

EXAMPLE

Vocabulary

Vocabulary

PRACTICE

Vocabulary

The absolute value of a number is always positive.

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Use a number line to fi nd the sum.

a. 4 1 (27) b. 22 1 3

Solution:

a.

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

Move 7 units to the left. End at 3. Start at 4. b.

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

Move 3 units to the right. Start at 2. End at 1.

4 1 (27) 5 23 22 1 3 5 1

Use a number line to fi nd the sum.

13. 21 1 (24) 14. 2 1 (28) 15. 25 1 9

16. 27 1 1 17. 0 1 (22) 18. 23 1 (26)

Use the rules of real number addition to fi nd the sum.

a. 219 1 (221) b. 2 1}2 1

3}2 c. 22.8 1 1.5

Solution:

If two numbers have the same sign, add their absolute values. The sum has the same sign as the numbers added.

a. 219 1 (221) 5 2(Z19Z 1 Z21Z) 5 2(19 1 21) 5 240

If two numbers have different signs, subtract the absolute value of the smaller number from the absolute value of the larger number. The sum has the same sign as the number with the larger absolute value.

b. 2 1}2 1

3}2 5 ) 3}2 ) 2 )2

1}2 ) 5

3}2 2

1}2 5 1

c. 22.8 1 1.5 5 2(Z22.8Z 2 Z1.5Z) 5 2(2.8 2 1.5) 5 21.3

Use the rules of real number addition to fi nd the sum.

19. 6.4 1 (20.3) 20. 8 1 (210) 21. 2100 1 (234)

22. s 1 2 2}3 1 4

3}4 23. 216 1 5

1}4 24. 55.1 1 (247.7)

4. Subtract Real NumbersDifference The result of subtracting one real number from another real number.

Find the difference.

a. 5 2 18 b. 26 2 9 c. 28 2 (23)

Solution:

To subtract b from a, add a and the opposite of b.

a. 5 2 18 5 5 1 (218) b. 26 2 9 5 26 1 (29) c. 28 2 (23) 5 28 1 3

5 213 5 215 5 25

PRACTICE

EXAMPLE

PRACTICE

Vocabulary

You can fi nd the sum of three or more numbers together by fi rst adding two of the numbers and then adding the result to the third.

EXAMPLE

Use grouping symbols around negative numbers in your work to keep track of signs as you simplify expressions.

EXAMPLE

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Find the difference.

25. 22.8 2 0.7 26. 1.9 2 21.1 27. 234 2 57

28. 3 3}5 2 (24) 29. 2

24}5 2 12

16}15

2 30. 73 2 (282)

5. Multiply Real NumbersProduct The result of multiplying two or more real numbers.

Find the product.

a. 21.7(4) b. 2 4}5 (210) c. 2(23)(28)

Solution:

The product of two numbers with the same sign is positive and the product of two numbers with different signs is negative.

a. 21.7(4) 5 26.8 b. 2 4}5 (210) 5

40}5 c. 2(23)(28) 5 [2(23)](28)

5 8 5 26(28)

5 48

Find the product.

31. 10(23) 32. 2 11}4 (6) 33. 2

5}8 12

24}15

2 12 25}47

234. 212(4)(23) 35. 2.5(10.4)(27) 36. 22.4(29.1)

Multiplicative inverse of a real number a The reciprocal of a, or 1}a. The product of a

and its multiplicative inverse is 1.

Find the multiplicative inverse of a.

a. a 5 9 b. a 5 24 c. a 5 2 1}8

Solution:

a. 1}a 5

1}9 b.

1}a 5

1}24

5 2 1}4 c.

1}a 5

1}2

1}8

5 2 8}1 5 28

Find the multiplicative inverse of the number.

37. 26 38. 1 39. 3}4

40. 2 7}5 41. 9

1}2 42. 25

15}32

EXAMPLE

PRACTICE

To multiply three or more real numbers, fi rst multiply two of the numbers, then multiply the result with the third number.

Vocabulary

PRACTICE

Vocabulary

You can check your answer by multiplying the original number by its inverse and making sure the product is 1.

PRACTICE

EXAMPLE

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6. Divide Real NumbersQuotient The result dividing a real number by another real number.

Find the quotient.

a. 35 4 (27) b. 226 4 (213) c. 2}3 4 12

12}18 2

Solution:

a. 35 4 (27) 5 35 12 1}7 2 b. 226 4 (213) 5 226 12

1}13 2

5 2 35}7 5

26}13

5 25 5 2

c. 2}3 4 12

12}18 2 5

2}3 12

18}12 2

5 2 36}36

5 21

Find the quotient.

43. 292 4 (24) 44. 22 1}4 4

5}8 45. 9 4

1}9

46. 1 4 12 5}2 2 47. 2

32}15

4 (28) 48. 26 2}3 4 10

4}9

7. Find Square RootsSquare root of a If b2 5 a, then b is the square root of a. Every positive nonzero real number a has two square roots, 2Ï

}a and Ï}a .

Radicand The number or expression inside a radical symbol.


a. 6Ï}

49 b. Ï}

1 c. 2Ï}

144

Solution:

a. 67 b. 1 c. 212


49. 2Ï}

400 50. 6Ï}

9 51. Ï}

81

52. Ï}

0 53. Ï}

4 54. 6Ï}

900

PRACTICE

EXAMPLE

The symbol 6in front of a number refers to the number and its opposite. For example, “66”is the same as “6 and 26.”

PRACTICE

Vocabulary

Vocabulary

EXAMPLE

Division by 0 is undefi ned, because 0 does not have a multiplicative inverse.

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QuizFor the given value of the variable, fi nd the opposite, absolute value, and multiplicative inverse.

1. a 5 216 2. y 5 7 3

}10

3. r 5 20.3


4. 51 2 (265) 5. 2 5}7

21}40

6. 6 5}9 4 121

2}3 2

7. 8 1 (215) 8. 22(235) 9. Ï}

64

10. 218 4 12}5 11. 6Ï

}

81 12. 22.3 2 4.9

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E. Properties and Real Numbers (pp. 15–18)

Taken together, the rational and irrational numbers make up the set of real numbers. The following examples illustrate some characteristics and properties of real numbers.

1. Classify Real NumbersWhole numbers A subset of the real numbers; A whole number is either 0 or one of the “counting numbers,” 1, 2, 3, . . .

Integers A subset of the real numbers; The integers are the set of whole numbers and their opposites, . . . 23, 22, 21, 0, 1, 2, 3, . . .

Rational numbers A subset of the real numbers; A rational number can be expressed as the ratio of two integers, and its decimal form terminates or repeats.

Irrational numbers A subset of the real numbers; An irrational number cannot be expressed as the ratio of two integers, and its decimal form neither terminates nor repeats.

Choose the word that best describes each: whole, integer, rational, or irrational.

a. 28 b. 3}4 c. 2Ï

}7

d. 20.1 e. Ï}

100 f. 4 1}3

Solution:

a. Integer; opposite of a b. Rational; ratio of two c. Irrational; cannot be whole number integers written as ratio of two integers nor as a terminating or repeating decimal

d. Rational; terminating e. Whole; Ï}

100 5 10 f. Rational; 13}3

5 4.333,

decimal a repeating decimal

Choose the word that best describes each: whole, integer, rational, or irrational.

1. 227 2. 22}7

3. 26 5}9

4. Ï}

64 5. 213.2 6. 0

Vocabulary

Use a Venndiagram to helpremember whichnumbers are partof other numbers.

PRACTICE

EXAMPLE


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Remember that theadditive inverseof a, 2a, is notnecessarily anegative number.

2. Order Real NumbersTo order real numbers from least to greatest, graph them first. Then read the numbers from left to right.

425

Solution:

45

2345

5 4.5 4

6 1 0 1 2 3 4 5 6

In order from least to greatest, the numbers are 25, 24.5, 2Ï}

4, and 24}5 .

Order each group of numbers from least to greatest.

7. 1.23, 1 2}3,

3}2, Ï

}

3 8. 0.08, 21.9, 2Ï}

0.04 ,Ï}

2 9. 6.01, Ï}

6 , 6 1}6, 6.1

3. Identify Properties of AdditionAdditive identity The number 0 is the additive identity. When 0 is added to a real number a, the sum equals a.

Additive inverse The opposite of a real number a is its additive inverse. The sum of a real number and its additive inverse always equals 0.

Identify the property of addition being illustrated.

a. (24 1 b) 1 3 5 24 1 (b 1 3) b. 7 1 x 5 x 1 7

c. 0 1 2 5}8

5 2 5}8

d. 17.3 1 (217.3) 5 0

Solution:

a. Associative; b. Commutative; c. Identity; d. Inverse; a changing the changing the adding 0 to a number plus grouping does order does not number does its opposite not change change the sum not change equals 0 the sum the number

10. 2k 1 0 5 2k 11. 7}8

1 128}7 2 5 2

8}7 1

7}8

12. 0 1 (21 1 n) 5 (0 1 (21)) 1 n 13. 0 5 25.6 1 5.6

14. r 1 (s 1 t) 5 (r 1 s) 1 t 15. 3 1 5 1 7 5 7 1 5 1 3

EXAMPLE

To order numbers,it is sometimeshelpful to writedecimalapproximations ofrational andirrationalnumbers.

PRACTICE

Vocabulary

EXAMPLE

PRACTICE


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Identify the property of addition being illustrated.

Order the numbers from least to greatest: 2 , 24.5, 25, 2 4.

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Remember thatassociative isrelated togrouping andcommutative isrelated to order.

PRACTICE

4. Identify Properties of MultiplicationMultiplicative identity The number 1 is the multiplicative identity. The product of a real number a and 1 equals a.

Identify the property of multiplication being illustrated.

a. 0 w 5 0 b. 56 (92 11) 5 (56 92) 11

c. g h 5 h g d. 24,978 1 5 24,978

e. 277 (21) 5 77 f. (r s) t 5 r (s t)

Solution:

a. Zero; a number b. Associative; changing c. Commutative; changing times 0 equals 0 the grouping does not the order does not change the product change the product

d. Identity; multiplying e. Negative one; the f. Associative; changing a number by 1 does product of a number the grouping does not not change the number and 21 is the opposite change the product of the number

Identify the property of multiplication being illustrated.

16. 6 p 5 p 6 17. 247 5 47 (21) 18. (25a)b 5 25(ab)

19. 235 5 1 (235) 20. 200 0 5 0 21. 21 (29) 5 9

5. Apply the Distributive PropertyDistributive property The product of two factors, where one factor is a sum, is the sum of the product of the first factor times the first addend plus the product of the first factor times the second addend. For example, a(b 1 c) 5 ab 1 ac.

Equivalent expressions Expressions that are equal in value, for any value of the variable.

Use the distributive property to write an equivalent expression.

a. 4(9 1 2) b. 8(b 2 3) c. 22n(n 1 6)

Solution:

a. 4(9 1 2) 5 (4)(9) 1 (4)(2) b. 8(b 2 3) 5 8[b 1 (23)] 5 36 1 8 5 8b 1 8(23) 5 44 5 8b 1 (224) 5 8b 2 24c. 22n(n 1 6) 5 (22n)(n) 1 (22n)(6)

5 22n2 1 (212n) 5 22n2 2 12n


22. 27(4 1 v) 23. c(2c 1) 24. 5m( 3 m)

25. 9(t 8) 26. 2}5 (15 20r) 27. 3p( 5 7p)


PRACTICE

Vocabulary

EXAMPLE

Vocabulary

EXAMPLE

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QuizChoose the word that best describes each number in the list. Then,write the numbers in order from least to greatest.

1. Ï}

8 , 4.1, 3, 8}3

2. 29.1, 29.02, 22}9 , 2Ï

}

9

3. 0}1, 20.01, 21.1, 2Ï

}

1

Identify the property being illustrated.

4. ( 8 3b) 2 8 (3b 2) 5. 7q 7q( 1)

6. 3}7 t

3}7 t 0 7. 1( 3x2) 3x2

8. 15(6 s) 90 15s 9. 12 h h 12

10. 6y(2) 2(6y) 11. 18(jk) 18j(k)

12. 6c(1 3c) 6c 18c2 13. 28m(0) 0

14. 3(2z 1) 6z 3 15. 37w 0 37w


16. 5(h 2) 17. 1}2

d(8 6d)

18. 8f( 4f 10)


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