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Rayhana J.

Variables, Expressions, and Properties

A __Variable__ is a symbol, usually a letter, to represent a number. Example: 4xn. That is called an **algebraic expression**, and when you replace **n **with a number (ex. 10), it becomes a __numerical expression__.

When you ** evaluate **an expression, you find it's numerical value. To avoid confusion, mathmeticians have agreed on a set of rules called

**P**arentheses

**E**xpressions

**M**ultiplication

**D**ivision

**A**ddition

**S**ubtraction

**PEMDAS**

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The problem is: 6-(5-3)+10. First, do the parenthesis.

5-3=2. So it becomes: 6-2+10. Then do: 6-2, and it will become 4. So now it is: 4+10. Add. 4+10=14.

Remember, some problems will have all 6 operations. Just follow **PEMDAS **for the right answer

Evaluate an Algebraic Expresion

If **a**=5, **b**=4, and **c**=8

4a-3b+1

This is actually: 4(5)- 3(4)+1 or 4x5-3x4+1

Evaluate 4x5 and 3x4. 20 and 12. Subtract. 20-12= 8. Add the 1. 9. The final answer is nine (9).

Note: N(N) means multiply. (N is a variable for any number)

Equations

A mathematical sentence that contains an **equal **(=) sign is called an ** equation**.

Ex. 7+8=15 3(6)= 18 x+2=5 (x is a **variable**)

Commutative Property

**Properties** are open sentences that are true for any numbers.

**Commutative Property **looks like this.

__Algebra__

a+b=b+a

__Arithmetic__

6+1=1+6

Associative Property

__Algebra__

a+(b+c)=(a+b)+c

**or**

a*(b*c)=(a*b)*c

__Arithmetic__

2+(3+8)=(2+3)+8

**or**

3*(4*5)=(3*4)*5

Distributive Property

__Algebra__

a+(b+c)= a(b)+a(c) (Remember, that means multiply)

**or**

a(b-c)=a(b)-a(c)

__Arithmetic__

4(6+2)=4*6+4*2

3(7-5)= 3*7-3*5

Identity Property

__Algebra__

a+0=a

**or**

a*1=a

__Arithmetic__

9+0=9

5*1=5

Transitive Property

__Algebra__

If a=b and b=c, then a=c also.

__Artithmetic__

If 2+4=6 and 6=3*2, then 3+4=3*2

Counterexample

A **counterexample **is an example that shows that a conjecture is false

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** Negative numbers- **any number less than 0.

Negative numbers like -86 and positive numbers like +125 and zero are members of the set of ** intergers**. Integers can be represented as points on the number line.

Writing Integers

**a 15 yard loss **= **-15**

**3 inches above normal **= **+3**

__Try it Out__

**a gain of 2 dollars****10 degrees below zero**

Comparing Integers

Using the greater than less than signs,(>) (<) compare the integers.

- Ex. 1 > -6
- -4 < -2

Try your own

- -5 and 4
- -3 and 2
- -1 and 1

** absolute value **is the distance the number is from 0 on the number line.

The ** absolute value** sign is a bar on each side of the number it looks ike the picture on the left.

Note: the **x **is a ** variable **that stands for any number. Even with negatives, the absolute value will look like that. Ex. -2 absolute value is 2.

Adding Integers

Find: -4+(-2). Using a number line is a great way to solve these problems. Starting from -4, go two units to the left (since it's a negative). The answer will be -6.

Remember, a negative plus a negative is always a negative.

Subtracting Integers

To subtract an integer, add its opposite or additive inverse. Ex. 4-7 is the same as 4+(-7), which is -3.

- 9-12
- -6-8
- 7-(-15)

Practice some of these problems in your book.

Multiplying Integers with Different Signs

6(-8)=-48. Just multiply regularly and add the negative sign. A negative times a positive is always a negative.

- -9(2)
- 5(-3)
- -8(6)

Multiplying Integers with the Same Sign

-4(-3)= 12. A negative times a negative equals a positive.

**Pratice**

- -3(-7)
- 6(4)
- -5
^{2}

Writing an Algebraic Expression

Five years older than her brother

Five years (5) older (+) her brother's age (a)

So the expression will be: 5+a

Write an Algebraic Equation

a number less than 8 is 22

a number (n) less than 8 (-8) is 22 (=22)

So the equaton will be n-8=22

Solving Addition Equations

x+3=7. We know that a number added to 3 will be 7. So to solve for x, we will subtract 3 from 7.

7-3=4. So x=4.

Solving Subtraction Equations

x-8= -3. Add 8 and -3 together. 8+-3=5. So x=5.

Solving Multiplication Equations

Solve 35d= 210

Divide each side by 35

35d/ 35 and 210/35

35/35=1 and 210/35= 6

1d=6. d=6

**Practice**

- 8x=72
- -4n=28

Solving Division Equations

Solve a/-3=-7.

Multiply each side by -3.

-3*-3=9 (this gets crossed out). -7*-3=21.

a=21

The End of Chapter 1

This is the end of Chapter 1. For extra practice, refer to the problems in your math book. Study the Flash Cards and take the Quiz if you want :). I wish you the best on the OAA's!

Chapter 2 coming soon.

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