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**Lesson 2 Operations with Rational and Irrational Numbers**

NCSCOS Obj.: 1.01; 1.02 Objective TLW State the coordinate of a point on a number line TLW Graph integers on a number line. TLW Add and Subtract Rational numbers. TLW Multiply and Divide Rational numbers

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**The Number Line Integers = {…, -2, -1, 0, 1, 2, …}**

-5 5 Integers = {…, -2, -1, 0, 1, 2, …} Whole Numbers = {0, 1, 2, …} Natural Numbers = {1, 2, 3, …}

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**SUBTRACT and use the sign of the larger number.**

Addition Rule 1) When the signs are the same, ADD and keep the sign. (-2) + (-4) = -6 2) When the signs are different, SUBTRACT and use the sign of the larger number. (-2) + 4 = 2 2 + (-4) = -2

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Karaoke Time! Addition Rule: Sung to the tune of “Row, row, row, your boat” Same signs add and keep, different signs subtract, keep the sign of the higher number, then it will be exact!

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= ? -4 -2 2 4 Answer Now

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-6 + (-3) = ? -9 -3 3 9 Answer Now

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**The additive inverses (or opposites) of two numbers add to equal zero.**

Example: The additive inverse of 3 is -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems.

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**What’s the difference between 7 - 3 and 7 + (-3) ?**

The only difference is that is a subtraction problem and 7 + (-3) is an addition problem. “SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.” (Keep-change-change)

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When subtracting, change the subtraction to adding the opposite (keep-change-change) and then follow your addition rule. Example #1: (-7) - 4 + (+7) Diff. Signs --> Subtract and use larger sign. 3 Example #2: - 3 + (-7) Same Signs --> Add and keep the sign. -10

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**11b + (+2b) Same Signs --> Add and keep the sign. 13b**

Okay, here’s one with a variable! Example #3: 11b - (-2b) 11b + (+2b) Same Signs --> Add and keep the sign. 13b

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**Which is equivalent to -12 – (-3)?**

12 + 3 12 - 3 Answer Now

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7 – (-2) = ? -9 -5 5 9 Answer Now

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Review 1) If the problem is addition, follow your addition rule. 2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.

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**Absolute Value of a number is the distance from zero.**

Distance can NEVER be negative! The symbol is |a|, where a is any number.

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Examples 7 = 7 10 = 10 -100 = 100 5 - 8 = -3= 3

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|7| – |-2| = ? -9 -5 5 9 Answer Now

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|-4 – (-3)| = ? -1 1 7 Purple Answer Now

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**Line up the decimals and add (same signs). -2.564**

Find the sum. 1) (-0.26) Line up the decimals and add (same signs). -2.564 2) Get a common denominator and subtract.

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**Find the difference. 3) Change subtraction to adding the opposite.**

Get a common denominator. Subtract and keep sign of the larger number.

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**Find the difference. 4) Change subtraction to adding the opposite.**

Get a common denominator and subtract.

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5) Solve 6.32 – y if y = -3.42 Substitute for y: (-3.42) 9.74

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Find the solution 6.5 – 9.3 = ? -3.2 -2.8 2.8 3.2 Answer Now

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Find the solution . Answer Now

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**A rational number is a number**

that can be written as a fraction. How can these be written as a fraction? 3 =

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Inequality Symbols

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**Ordering Rational Numbers**

2 ways to order from least to greatest Get a common denominator Change the fractions to decimals (numerator demoninator)

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**Which rational number is bigger?**

1) Get a common denominator. 2) or convert the fraction to a decimal. < 4 1 < 3 8

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**Which rational number is bigger?**

7 4 or 1 6 Get a common denominator or convert to a decimal 1.75 < 1.83 42 24 < 44

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**Which symbol makes this true?**

5 7 __ 3 4 < > = Answer Now

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**Which symbol makes this true?**

-2 -1 __ 9 4 < > = Answer Now

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Multiplying Rules 1) If the numbers have the same signs then the product is positive. (-7) • (-4) = 28 2) If the numbers have different signs then the product is negative. (-7) • 4 = -28

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**Multiplying fractions:**

Examples 1) (3x)(-8y) -24xy 2) Write both numbers as a fraction. Cross-cancel if possible. Multiplying fractions: top # • top # Bottom # • bottom# = -16

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**When multiplying two negative numbers, the product is negative.**

True False Answer Now

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**When multiplying a negative number and a positive number, use the sign of the larger number.**

True False Answer Now

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3) = =

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**Multiply: (-3)(4)(-2)(-3)**

72 -72 36 -36 Answer Now

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**an easy way to determine the sign of the answer**

When you have an odd number of negatives, the answer is negative. When you have an even number of negatives, the answer is positive. 4) (-2)(-8)(3)(-10) Do you have an even or odd number of negative signs? 3 negative signs -> Odd -> answer is negative -480

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**Positive or negative answer? Positive - even # of negative signs (4)**

Last one! 5) Positive or negative answer? Positive - even # of negative signs (4) Write all numbers as fractions and multiply. =12

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**What is the sign of the product of (-3)(-4)(-5)(0)(-1)(-6)(-91)?**

Positive Negative Zero Huh? Answer Now

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Dividing Rules 1) If the numbers have the same signs then the quotient is positive. -32 ÷ (-8 )= 4 2) If the numbers have different signs then the quotient is negative. 81 ÷ (-9) = -9

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**When dividing two negative numbers, the quotient is positive.**

True False Answer Now

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**When dividing a negative number and a positive number, use the sign of the larger number.**

True False Answer Now

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**The reciprocal of a number is called its multiplicative inverse.**

The reciprocal of is where a and b 0. The reciprocal of a number is called its multiplicative inverse. A number multiplied by its reciprocal/multiplicative inverse is ALWAYS equal to 1.

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Example #1 The reciprocal of is Example #2 The reciprocal of -3 is

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**Basically, you are flipping the fraction!**

We will use the multiplicative inverses for dividing fractions.

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**Which statement is false about reciprocals?**

Reciprocals are also called additive inverses A number and its reciprocal have same signs If you flip a number, you get the reciprocal The product of a number and its reciprocal is 1 Answer Now

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Examples 1) When dividing fractions, change division to multiplying by the reciprocal.

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2)

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**What is the quotient of -21 ÷ -3?**

18 -18 7 -7 Answer Now

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. Answer Now

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