name date period lesson 1 reteachmwillmarth.org/wp-content/uploads/2018/11/m7a_reteach.pdfdivide...

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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 1 ReteachIntegers and Absolute Value

Example 1 Write an integer for each situation. Then identify its opposite and describe what it means.

a. 16 feet below the surface b. 5 strokes over par

The integer is -16. The integer is +5 or 5.The opposite is 16. The opposite is -5. It means 16 feet above the surface. It means 5 strokes below par.

Numbers on opposite sides of zero and the same distance from zero have the same absolute value.

The symbol for absolute value is two vertical bars on either side of the number. |2| = 2 and |-2| = 2

Example 2 Evaluate each expression.

a. |-4| b. |-3| + |6|

|-4| = 4 On the number line, -4 is 4 units from 0.

Exercises Write an integer for each situation. Then identify its opposite and describe what it means. 1. 2 inches less than normal 2. 13°F above average

3. a deposit of $50 4. a loss of 8 yards

Evaluate each expression if x = 8 and y = - 3. 5. 12 + |y| 15 6. x - |y| 5 7. 2|x| + 3|y| 25

8. x + |y| 11 9. 6|y| 18 10. 3x - 4|y| 12

A negative number is a number less than zero. A positive number is a number greater than zero. The set of integers can be written {…, -3, -2, -1, 0, 1, 2, 3, …} where … means continues indefinitely. Two integers can be compared using an inequality, which is a mathematical sentence containing < or >.

|- 3| + |6| = 3 + 6 = 9

6-6 420-4 -2

4 units |-3| = 3, |6| = 6

Simplify.

- 2; +2 or 2; 2 inches more than normal

+ 50 or 50; -50; a withdrawal of $50

+ 13 or 13; -13; 13°F below average

- 8; +8 or 8; a gain of 8 yards

Program: Pre-Algebra

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Math Accelerated • Chapter 2 Operations with Integers

001_005_AM_ANC_C02_L1_664489.indd 7/21/12 12:39 AM s-74user001_005_AM_ANC_C02_L1_664489.indd Page 1 7/21/12 12:39 AM s-74user /Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES.../Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES...

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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 2 ReteachAdding Integers

Add Integers

with the Same Sign

Add their absolute values. The sum is:

· positive if both integers are positive.

· negative if both integers are negative.

Example 1 Find the sum -3 + (-4).-3 + (-4) = -7 Add ⎪-3⎥ and ⎪-4⎥ . The sum is negative.

Add Integers

with Different Signs

Subtract their absolute values. The sum is:

· positive if the positive integer’s absolute value is greater.

· negative if the negative integer’s absolute value is greater.

Example 2 Find the sum -5 + 4.-5 + 4 = ⎪-5⎥ - ⎪4⎥ Subtract ⎪4⎥ from ⎪-5⎥ .

= 5 - 4 or 1 Simplify.

= -1 The sum is negative because ⎪-5⎥ > ⎪4⎥ .

Two numbers with the same absolute value but different signs are opposites. An integer and its opposite are also called additive inverses. This property is useful when adding 2 or more integers.

Example 3 Find the sum 12 + (- 4) + 9 + (-7).12 + (- 4) + 9 + (-7) = 12 + 9 + (- 4) + (-7) Commutative Property

= (12 + 9) + [- 4 + (-7)] Associative Property

= 21 + (-11) or 10 Simplify.

Exercises Find each sum. 1. 6 + (-3) 2. -3 + (-5) 3. 7 + (-3)

4. -4 + (-4) 5. -8 + 5 6. -12 + (-10)

7. 7 + (-18) 8. -12 + (-15) 9. 10 + (-14)

10. -33 + 19 11. -20 + (-5) 12. -15 + (-20)

13. -15 + 4 14. -34 + 29 15. 46 + (-32)

16. 6 + (-14) + (-5) + (-6) 17. -18 + 9 + (-7) + 18

18. 5 + 13 + (-11) + 6 19. -20 + 15 + (- 10) + 3

20. -33 + (-7) + 20 + 4 21. 16 + (-12) + 21 + (-25)

3 -8 4

-8 -3 -22

-11 -27 -4

-14

-11

-19

13

-16

-25 -35

14-5

2

-12

0


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 3 ReteachSubtracting Integers

Subtract

IntegersTo subtract an integer, add its additive inverse.

Example 1 Find each difference.

a. 9 - 17 b. -7 - 3 9 - 17 = 9 + (-17) To subtract 17, add -17. -7 - 3 = -7 + (-3) To subtract 3, add -3.

= -8 Simplify. = -10 Simplify.

Example 2 Find each difference.

a. 4 - (-5) b. -6 - (-2) 4 - (-5) = 4 + 5 To subtract -5, add 5. -6 - (-2) = -6 + 2 To subtract -2, add 2.

= 9 Simplify. = -4 Simplify.

To find the distance between two integers on a number line, you can count the units on the number line or use absolute value.

Example 3 Find the distance between 4 and -9 on a number line. ⎪4 - (-9)⎥ = ⎪13⎥ Find the absolute value of the difference of 4 and -9.

= 13 Simplify.

Exercises Find each difference. 1. 9 - 16 -7 2. 7 - 19 -12 3. 12 - 21 -9

4. -5 - 3 -8 5. -8 - 9 -17 6. -13 - 17 -30

7. -24 - 8 -32 8. 18 - (-9) 27 9. 26 - 49 -23

10. -45 - (-26) -19 11. -15 - (-25) 10 12. 29 - (-6) 35

Find the distance between the integers on a number line. 13. -2 and -6 4 14. 9 and -9 18 15. 0 and -5 5

16. -12 and 15 27 17. -1 and -11 10 18. -4 and -16 12


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 4 ReteachMultiplying Integers

Multiply Integers

with Different SignsThe product of two integers with different signs is negative.

Example 1 Find each product.

a. 4(-3) b. -8(5) 4(-3) = -12 -8(5) = -40

Multiply Integers

with the Same SignThe product of two integers with the same sign is positive.

Example 2 Find each product.

a. 6(6) b. -7(-4) 6(6) = 36 -7(-4) = 28

Example 3 Evaluate 4xy if x = 3 and y = -5.4xy = 4(3)(-5) Replace x with 3 and y with -5.

= [4(3)](-5) Associative Property of Multiplication

= 12(-5) The product of 4 and 3 is positive.

= -60 The product of 12 and -5 is negative.

Exercises Find each product.

1. -5(7) -35 2. 6(-9) -54 3. -10 · 4 -40

4. -12 · -2 24 5. 5(-11) -55 6. -15(-4) 60

7. 11(3)(-2) -66 8. -5(-6)(7) 210 9. -2(-5)(-9) -90

Evaluate each expression if x = -4 and y = 8.

10. 4x -16 11. 3y 24 12. -12x 48

13. -2xy 64 14. 5xy -160 15. -3x(-y) -96


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 5 ReteachDividing Integers

Divide Integers

with Different SignsThe quotient of two integers with different signs is negative.

Example 1 Find each quotient.

a. 36 ÷ (-4) The signs are different. 36 ÷ (-4) = -9 The quotient is negative.

b. - 42 − 6 The signs are different.

- 42 − 6 = -7 The quotient is negative.

Divide Integers

with the Same SignThe quotient of two integers with the same sign is positive.

Example 2 Find each quotient.

a. 14 ÷ 2 The signs are the same.

14 ÷ 2 = 7 The quotient is positive.

b. -25 − -5

-25 − -5 = -25 ÷ (-5) The signs are the same.

= 5 The quotient is positive.

To find the mean, or average, of a set of numbers, find the sum of the numbers and then divide by the number of items in the set.

Example 3 The diving depths in feet of 7 scuba divers studying schools of fish were -12, -9, -15, -8, -20, -17, and -10. Find the mean diving depth.

-12 + (-9) + (-15) + (-8) + (-20) + (-17) + (-10) −−− 7 = -91 − 7 Find the sum of the diving depths.

Divide by the number of divers.

= -13 Simplify.

The mean diving depth is -13 feet, or 13 feet below sea level.

Exercises Find each quotient.

1. 40 ÷ (-5) 2. -18 ÷ (-2) 3. -24 ÷ 6

4. -28 − 2 5. 36 − -4 6. -150 −

-25

7. The low temperatures in degrees Fahrenheit for a week were -3, 5, -9, 2, 6, -11, and -4. Find the mean temperature.

8. During 5 rounds of golf, James had scores of 2, -1, 0, -2, and -4. Find the mean of his golf scores.

-2°F

-1

-8 9 -4

-14 -9 6


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 6 ReteachGraphing in Four Quadrants

Example 1 Graph and label each point on a coordinate plane. Name the quadrant in which each point lies.

a. M(-2, 5) Start at the origin. Move 2 units left. Then move 5 units up and draw a dot. Point M(-2, 5) is in Quadrant II.

b. N(4, -4) Start at the origin. Move 4 units right. Then move 4 units down and draw a dot. Point N(4, -4) is in Quadrant IV.

Exercises Graph and label each point on the coordinate plane. Name the quadrant in which each point is located.

1. A(2, 6) I 2. B(-1, 4) II

3. C(0, -5) none 4. D(-4, -3) III

5. E(2, 0) none 6. F(3, -2) IV

7. G(-4, 4) II 8. H(2, -5) IV

9. I(6, 3) I 10. J(-5, -8) III

11. K(3, -5) IV 12. L(-7, -3) III

The coordinates are (negative, positive).

The coordinates are (negative, negative).

The coordinates are (positive, positive).

The coordinates are (positive, negative).

y

O1-1-2-3-4

234

1

-1-2-3-4

2 3 4 x

(+, +)(-, +)

(-, -) (+, -)

M

N

x

y

O

1-1-2-3-4

2345

1

-1-2-3-4

2 3

2 x

y

O1-1-3-2-4-5-6-7-8

34

12

5678

-1-2-3-4-5-6-7-8

3 4 5 6 7 8

A

E

B

F

KHC

L D

GI

J


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 1 ReteachFractions and Decimals

Example 3 Replace � with < , > , or = to make -0.37 � - 4 − 11 a true sentence.

-0.37 � - 4 − 11

-0.37 � -0. −− 36 Write 4 −

11 as a decimal.

-0.37 < -0. −− 36 -0.37 is to the left of -0. −−

36 on the

number line, so -0.37 < -0. −− 36 .

Exercises Write each fraction as a decimal. Use a bar to show a repeating decimal.

1. 7 − 20 0.35 2. 2 − 11 0. −−

18 3. 5 − 9 0. −

5

4. - 7 − 9 -0. −

7 5. 27 − 40 0.675 6. - 2 − 3 -0. −

6

Replace each � with < , > , or = to make a true sentence.

7. 5 − 8 � 6 − 9 < 8. 4 − 5 � 0.8 = 9. 7 − 8 � 4 − 5 >

10. 0.16 � 4 − 25 = 11. - 11 − 40 � -0.02 < 12. 7 − 8 � 0.88 <

Some fractions can be written as decimals by making equivalent fractions with denominators of 10, 100, or 1,000. All fractions can be written as decimals by dividing the numerator by the denominator. Repeating decimals have a pattern in their digits that repeats without ending. If the repeating digit is zero, then the decimal is a terminating decimal.

It may be easier to compare numbers when they are written as decimals.

Example 1 Write 3 − 8 as a decimal.

3 − 8 0.375 8 � �� 3.000 2 4 60 56 40 40 0

0.375 is a terminating decimal.

Example 2 Write 4 − 9 as a decimal.

4 − 9 0.444 9 � �� 4.000 3 6 40 36 40 36 4

0.444... is a repeating decimal. You can indicate that a decimal repeats by writing a bar or line over the repeating digit(s): 4 − 9 = 0. − 4 .

-0.37 -0.36

-0.39 -0.37 -0.35 -0.33 -0.31-0.40 -0.38 -0.36 -0.34 -0.32 -0.30


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Math Accelerated • Chapter 3 Operations with Rational Numbers

001_005_AM_ANC_C03_L1_664489.indd Page 1 24/07/12 1:52 AM s-019001_005_AM_ANC_C03_L1_664489.indd Page 1 24/07/12 1:52 AM s-019 /Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES.../Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES...

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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 2 ReteachRational Numbers

Example 2 Identify all sets to which each number belongs.

a. -0.08 This is neither a whole number nor an integer.

Since -0.08 can be written as - 8 −

100 , it is rational.

b. 8.282282228... This is a nonterminating and nonrepeating decimal. So, it is irrational.

ExercisesWrite each number as a fraction.

1. 1 1 − 5 6

− 5

2. -2 - 2

− 1

3. 0.7 7

− 10

4. -9.08 -9 2

− 25

5. -0.0 − 6 - 1

− 15

6. 6 8 − 11 74

− 11

Identify all sets to which each number belongs. 7. -12 8. 8.5 9. 582 integer, rational rational natural, whole, integer,

rational

10. 0 11. -68 12. 1 − 5 whole, integer, rational integer, rational rational

A number that can be written as a fraction is a rational number. Mixed numbers, integers, terminating decimals, and repeating decimals can all be written as fractions. Any number that can be expressed as a −

b ,

where a and b are integers and b ≠ 0 is a rational number.

Numbers can be classified into a variety of different sets. The diagram at the right illustrates the relationships among the sets of natural numbers, whole numbers, integers, and rational numbers.

Decimal numbers such as π = 3.141592... and 6.767767776... are infinite and nonrepeating. They are called irrational numbers.

a. 3 2 − 5

3 2 − 5 = 17 − 5 Write the mixed number as an improper fraction.

b. 0.14

0.14 is 14 hundredths. 0.14 = 14 − 100 or 7 − 50 Simplify.

Example 1 Write each number as a fraction.

RationalNumbers

3.777...

-

-1.5Integers

Whole0

Natural7

-100 -8

13

5 78

12

34

6.32


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 3 ReteachMultiplying Rational Numbers

Example 1 Find 7 1 − 2 · 2 2 − 3 . Write in simplest form.7 1 − 2 · 2 2 − 3 = 15 − 2 · 8 − 3 Rename mixed numbers as improper fractions.

= 15 − 2 · 8 − 3 Divide 15 and 3 by 3, and 8 and 2 by 2.

= 5 · 4 − 1 · 1 Multiply.

= 20 − 1 or 20 Simplify.

To multiply fractions, multiply the numerators and multiply the denominators: a − b

· c − d

= a · c − b · d

, where

b, d ≠ 0. Fractions may be simplified before or after multiplying. For negative fractions, assign the negative sign to the numerator.

Algebraic expressions are expressions which contain one or more variables. Variables can represent fractions in algebraic expressions.

Example 2 Evaluate 2 − 3 ab if a = 3 3 − 7 and b = - 5 − 12 . Write the product in simplest form.

2 − 3 ab = 2 − 3 (3 3 − 7 ) (-

5 − 12 ) Replace a with 3 3 − 7

and b with - 5 −

12 .

= 2 − 3 ( 24 − 7 ) ( -5 − 12 ) Rename 3 3 −

7 as 24 −

7 .

= 2 − 3 ( 24 − 7 ) ( -5 − 12 ) The GCF of 24 and 12 is 12.

= - 20 − 21 or -

20 − 21 Simplify.

ExercisesFind each product. Write in simplest form.

1. 1 − 2 · 3 − 5 2. - 8 − 9 · 5 − 16 3. 4 − 5 · 5 − 8

4. 3 − 10 · (- 1 − 4 ) 5. -2 1 − 8 · (- 4 4 − 7 ) 6. 2 4 − 9 · (- 3 6 − 11 )

Evaluate each expression if x = 7 − 10 , y = -4 2 − 5 , and z = - 4 − 7 . Write the product in simplest form.

7. xy 8. yz 9. xyz

10. 5 5 − 6 xz 11. 2 − 5 (-x) 12. 9 − 10 y

5 4

2

1

3 −

10

9 5 − 7

-3 2 −

25

-2 1 − 3

- 7

− 25

-3 24 −

25

2 18 −

35 1 19

− 25

- 3

− 40

1 − 2

- 5

− 18

- 8 2 − 3


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 4 ReteachDividing Rational Numbers

Example 1 Find 3 − 4 ÷ 5 − 8 . Write in simplest form.

3 − 4 ÷ 5 − 8 = 3 − 4 · 8 − 5 Multiply by the multiplicative inverse of 5 − 8

, 8 − 5

.

= 3 − 4 · 8 − 5 Divide 4 and 8 by their GCF, 4.

= 6 − 5 or 1 1 − 5 Simplify.

Algebraic fractions are fractions which contain one or more variables. You can divide algebraic fractions just as you would divide numerical fractions.

Example 2 Find 4 − qrs ÷ 10 − qs . Write in simplest form.

4 − qrs ÷ 10 − qs = 4 − qrs · qs

− 10 Multiply by the reciprocal of 10 − qs , qs

− 10

.

= 4 − qrs · qs

− 10 Divide out common factors.

= 2 − 5r Simplify.

Exercises Find each quotient. Write in simplest form.

1. 5 − 16 ÷ 5 − 8 2. 7 − 9 ÷ 2 − 3 3. 16 − 21 ÷ (- 2 − 7 )

4. - 4 − 5 ÷ 3 − 10 5. 18 − 21 ÷ 3 6. -4 5 − 8 ÷ (-3 1 − 3 )

7. 2x − y ÷ 3 − y 8. c − 4d ÷ 3 − 8d 9. 4a − b ÷ 2ac − b

10. m − 9 ÷ mn 2 − 3 11. ab − 9 ÷ bc − 12 12. 2st − q ÷ 4t − q

13. 15yz

− 6x ÷ 10z − 3x 14. de − 20f ÷ e − 2f 15. 6i − 5gh ÷ 8i − 3h

2

1

5

1

Two numbers whose product is 1 are called multiplicative inverses or reciprocals.

For any fraction a − b

, where a, b ≠ 0, b − a is the multiplicative inverse and a − b

· b − a = 1.

This means that 2 − 3

and 3 − 2

are multiplicative inverses because 2 − 3

· 3 − 2

= 1.

To divide by a fraction, multiply by its multiplicative inverse: a − b

÷ c − d

= a − b

· d − c = ad − bc

, where b, c, d ≠ 0.

2

1 1

1 − 2

-2 2 − 3

-2 2 − 3

1 −

3 n 2

2x −

3 2c

− 3

2 − c

s − 2

9 −

20g

4a −

3c

1 1 − 6

1 31 −

80 2 −

7

3y

− 4

d −

10


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 5 ReteachAdding and Subtracting Like Fractions

Example 1 Find 3 − 8 + (- 7 − 8 ) . Write in simplest form.

3 − 8 + (- 7 − 8 ) = 3 + (-7) − 8 The denominators are the same. Add the numerators.

= -4 − 8 or - 1 − 2 Simplify.

Example 2 Find 1 2 − 9 + 3 4 − 9 . Write in simplest form.

1 2 − 9 + 3 4 − 9 = (1 + 3) + ( 2 − 9 + 4 − 9 )

= 4 + 2 + 4 − 9

= 4 6 − 9 or 4 2 − 3

To subtract fractions with like denominators, subtract the numerators and write the difference over the

denominator. So, a − c - b − c = a - b − c , where c ≠ 0.

Example 3 Find 3 − 8 - 5 − 8 . Write in simplest form.

3 − 8 - 5 − 8 = 3 - 5 − 8 The denominators are the same. Subtract the numerators.

= - 2 − 8 or - 1 − 4 Simplify.

ExercisesFind each sum. Write in simplest form.

1. 11 − 12 + 9 − 12 2. 13 − 15 + 9 − 15 3. 4 − 9 + 8 − 9

4. -6 7 − 12 + (-8 11 − 12 ) 5. -4 9 − 14 + 3 5 − 14 6. 2 3 − 5 + (- 1 − 5 )

Find each difference. Write in simplest form.

7. 19 − 20 - 17 − 20 8. 23 − 25 - 8 − 25 9. 5 − 9 - 2 − 9

10. 3 − 7 - 5 − 7 11. 4 − 12 - 7 − 12 12. 14 − 15 - 9 − 15

Add the whole numbers and fractions separately or write as

improper fractions.

Add the numerators.

Simplify.

To add fractions with the same denominators, called like denominators, add the numerators and write the sum over the denominator.

So, a − c + b − c = a + b − c , where c ≠ 0.

1 2

− 3

- 1

− 4

- 2

− 7

1

− 10

3

− 5

1 7

− 15

-1 2

− 7

1 1

− 3

1

− 3

2 2

− 5

1

− 3

-15 1

− 2


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NAME ___________________________________________________ DATE _____________________ PERIOD __________

Lesson 6 ReteachAdding and Subtracting Unlike Fractions

Example 1 Find 4 − 7 + 1 − 3 . Write in simplest form.

4 − 7 + 1 − 3 = 4 − 7 · 3 − 3 + 1 − 3 · 7 − 7 Use 7 · 3 or 21 as the common denominator.

= 12 − 21 + 7 − 21 Rename each fraction with the common denominator.

= 19 − 21 Add the numerators.

To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify.

Example 2 Find 9 2 − 9 - 8 5 − 6 . Write in simplest form.

9 2 − 9 - 8 5 − 6 = 83 − 9 - 53 − 6 Write the mixed numbers as improper fractions.

= 83 − 9 · 2 − 2 - 53 − 6 · 3 − 3 Rename fractions using the LCD, 18.

= 166 − 18 - 159 − 18 Simplify.

= 7 − 18 Subtract the numerators.

ExercisesFind each sum. Write in simplest form.

1. 8 − 9 + 2 − 5 2. - 2 − 3 + 1 − 4 3. 7 − 8 + 1 − 4

4. 7 4 − 9 + 9 1 − 6 5. -7 1 − 2 + (-3 2 − 9 ) 6. -10 1 − 7 + 6 1 − 4

Find each difference. Write in simplest form.

7. 3 − 8 - 1 − 12 8. - 7 − 9 - 4 − 5 9. 5 − 12 - (- 3 − 8 )

10. 5 1 − 10 - 3 2 − 3 11. -6 3 − 5 - (-2 1 − 4 ) 12. 10 5 − 6 - (-5 2 − 3 )

Fractions with different denominators are called unlike fractions. To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify.

1 13 −

45 - 5

− 12

1 1 − 8

16 11 −

18 -3 25

− 28

-10 13 −

18

-1 26 −

45

-4 7 −

20

19 −

24

16 1 − 2

7 −

24

1 13 −

30


Vendor: Aptara


Grade: AMPDF Pages



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