Integrated Algebra - Chapter 1

8/9/2019 Integrated Algebra - Chapter 1

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Integrated Algebra Chapter 1: Working with Numbers and Variables

Mr. Smith 1 Sachem High School North

Working with Numbers and Variables

Introduction Vocabulary

Constant: A quantity that does not change. Variable: A quantity that can change. Expression: 2 + 3x5y is an expression, notice the absence of an equals sign. Equation: 3 +x = 4y is an equation because one quantity equals another. Inequation: Opposite of an equation; where one expression cannot equal another. Set: A collection of objects or numbers. Elements: members of a set.

SetA with elements 2, 4, 6, 8, 10: = {2,4,6,8,10}

Subset: A set within a set. Domain: Also known as the replacement set, the values that can be represented byx.

Introduction to Inequalities

Inequalities can also be used to represent an interval.

The Use of Variables

Variables can be used in 3 general ways:

1. Placeholder for numbers2. As a variable, to represent quantities that change3. To represent quantities in a given formula.


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1.1The Set of Real Numbers

The Set of Real Numbers contains several subsets:

Natural Numbers (Counting Numbers)

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

The Image to keep in mind when thinking about the Set of Real Numbers:

REAL NUMBERS

IRRATIONAL NUMBERSRATIONAL NUMBERS

INTEGERS

WHOLE NUMBERS

NATURAL NUMBERS


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1.2Integers and Absolute Value

Signed Numbers: Numbers with a positive (+) or negative (-) sign.Keep in mind though, positive numbers are often written with no sign.

The relationship of allReal Numbers can be represented by a Number Line:

A Number Line extends indefinitely in opposite directions, centered at zero whichis known as theorigin.

Positive Integers are to the right of zero, and Negative Integers are to the left.

Absolute Value: The distance from any number on a number line to zero.

Therefore two numbers that are inverses have the same absolute value:


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1.3Addition

Representing Addition: Addition can be represented by the plus (+) sign.

The numbers being added are addends, and the solution is known as the sum.

Properties of Addition:

Commutative Property of Addition: when adding terms, order doesnt matter.

Associative Property of Addition: when grouping addends their sums do not change.

Additive Identity: Any number added to zero is that number.

Additive Inverse: When adding two numbers with identical signs the sum is zero.

Adding numbers with the same sign: Add the numbers, and keep the common sign.

Examples:

Adding numbers with different signs: Subtract and keep the sign of the larger number.

Examples:


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1.4Subtraction

Representing Subtraction: Subtraction can be represented by the minus () sign.

The Solution from a subtraction Problem is known as the difference.

Here are some examples:

16 5 =

1 11 =

6 7 =

8 9 =

2 13 =

1.5Multiplication

Multiplication is a little different, because there are several ways to indicate if numbers are being

multiplied.

Generally, people are most familiar with the standard multiplication sign (x).

Since, this can become a problem when using the variablex; there are two other ways toexpress theproduct (solution from multiplication) of numbers.

Use of Parentheses:

Centering a dot between two quantities:


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Also when multiplying a number (coefficient) and a variable, they can be written next toeach other and multiplication is implied.

Properties of Multiplication:

Commutative Property of Multiplication: when multiplying terms, order doesnt matter.

Associative Property of Multiplication: Regardless of grouping, the product of numbers is

always the same.

Distributive Property: When multiplying a sum by a number, it is equal to the sum of the

products between the number and each addend.

Multiplicative Inverse: When multiplying a number by its inverse the answer is always 1.

Zero Property: Any number multiplied by zero equals zero.

Multiplying numbers with the same sign: Multiply the numbers, the product is positive.

Multiplying numbers with different signs: Multiply the numbers, the product is negative.


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1.6Division

Representing Division: Can be done with a typical division sign, or as a fraction.

In fractional form, the numerator (top number) is known as the dividend, and the denominator

(bottom number)is called the divisor. The Solution to a division problem, is known as theQuotient.

The Golden Rules of Division

Dividing numbers with the same sign: Divide the numbers, the quotient is positive.

Dividing numbers with different signs: Divide the numbers, the quotient is negative.

I. Dividing by zero is not allowed.

II. Any number divided by itself is 1. (except zero)

III. Any number divided by one is that number.


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1.7 Exponents & Scientific Notation

An exponent (power) is the number of times that abase number is multiplied by itself.

For example: two raised to the sixth poweris represented as such:

In this example, Two is the base, And 6 is the exponentorpower

Which is the amount of times two is multiplied by itself.Now before we continue please note:

The exponent is the number of times the base is multiplied notthe number to multiply by.

There are two exponent values to remember:

Some number a raised to thefirst power:

Any number to the first power is itself.

Some numberz raised to thezero power:

Any number to the zero power is one.


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Properties of Positive Exponents:

1) Product RuleWhen multiplying values in exponential form with the same base, we can add the

exponents as such:

2) Quotient RuleDividing values in exponential form with the same base is the complete opposite,

we can subtract the exponents like so:

3) Power to a Power RuleTo raise a value that is already in exponential form to another power, multiply the

exponents.

Negative Exponents:

If a number has a negative exponent it really means the inverse of that Value.

Recall: the inverse of a number is 1 over that number.

But, what happens when there is a negative exponent in the denominator of a fraction?

The procedure remains the same: Take the inverse of the value, and solve.

Negative exponentthrow it in the basement.

Frank DeMeo, Math Teacher at Seneca


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Scientific Notation is the way we represent large and small numbers that include several zeros

in them.

Example 1: Earth is approximately 4.536 Billion km from the former planet, Pluto.

Now we can write that in scientific notation like so:

It is even more useful when describing small objects.

Example 2: The mass of an electron is:

Written in scientific notation it runs as such:

Now notice the difference between the two numbers weve generated.

Why is the ten raised to a positive power in the first example anda negative power in the latter?

Practice Problems:

How absurdly

simple!

Quite so,

Watson.


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1.8 Order of Operations

The Million Dollar Question: Where do we begin?

The Order of Operations are as follows: Do the operations inside the Parentheses, if any. Do any Exponents Multiply and/or Divide in order from left to right. Add and/or Subtract from left to right.

There are a few ways to remember this, but I prefer to use the Acronym: PEMDAS

Some people like remembering the mnemonic:

Please Excuse My Dear Aunt Sally

With that in mind, lets try our original problem:

The Correct

Answer is:

369


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1.9 The Language of Math

As for any foreign language in order to effectively communicate,

one must be able to translate and make sense of what is going on.

Luckily, unlike Spanish or Italian, we translate between English

and Math every day. Lets take a look:

Jack weighs 100 lbs. more than his brother Phil.

In 3 years, Tom will be halfhis moms age.

In my wallet, I have $7 less than I did yesterday.

Thats the general idea, now we get into the specifics.

How the expression is phrased will explain exactly what is going on. Here are some key words to

look for:

Addition Examples: Subtraction Examples:

The Sum of n and 5. 3subtracted fromx.

5 more than n 3 less thanx.

xdiminished by 3.

Multiplication Examples: Division Examples:

4timesx. ndivided by 2.

The product of4 andx. One-half ofn.

For English, press one.

Para Espaol, pulsar dos.

If your brother Johnny is

three years older than

your sister Mary thenplease stay on the line.


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The Language of Math: Inequalities

When it comes to translating between Math and English, probably the most useful and important

translations are inequality expressions.

Recall the four inequality symbols we discussed earlier:

There are a few phrases that can be represented by these symbols as well, here are a few:

The Symbolic Representation for,Less Than

She has less than $5 in her pocket.

*She hasat most $5 in her pocket.

*She hasno more than $5 in her pocket.

The Symbolic Representation for,Greater Than

He hasmore than $300 in the bank.

*He hasat least $300 in the bank.

*He hasno less than $300 in the bank.

*These expressions are most often misinterpreted incorrectly due to tricky wording.

When examining these expressions, try and visualize what is going on, I like to think of money

because we deal with it on a daily basis, and I know that if I have at most$3 in my pocket, I

should probably hit the atm before going to a restaurant so I have money to tip the waiter.

Integrated Algebra - Chapter 1

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