**Learning Objectives**

After completing this tutorial, you should be able to:

- Identify and use the addition and multiplication identity properties.
- Identify and use the addition and multiplication inverse properties.
- Identify and use the addition and multiplication commutative properties.
- Identify and use the addition and multiplication associative properties.
- Identify and use the distributive property.
- Know the key words that translate into an equal sign.
- Know the symbol for 'not equal to'.
- Know the symbol for and the meaning of 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to'.
- Write mathematical expressions that have an equal, less than, greater than, less than or equal to, or greater than or equal to sign.

** Introduction**

It is important to be familiar with the properties in this tutorial.
They lay the foundation that you need to work with equations, functions,
and formulas all of which are covered in later tutorials, as well as, your
algebra class.

We will start with the properties for real numbers and then look at writing out equalities and inequalities in mathematical statements.

** Tutorial**

*The additive identity is 0*

*a + 0 = 0 + a = a*

In other words, when you add 0 to any number, you end up with that
number as a result.

*Multiplication identity is 1*

*a(1) = 1(a) = a*

And when you multiply any number by 1, you wind up with that number
as your answer.

*For each real number a, there is a unique real number,
denoted -a,*

*such that *

*a + (-a) = 0.*

In other words, when you add a number to its additive inverse, the
result is 0. Other terms that are synonymous with additive inverse are
negative and opposite.

*For each real number a, except 0, there is a unique
real number such
that*

In other words, when you multiply a number by its multiplicative inverse
the result is 1. A more common term used to indicate a multiplicative
inverse is the **reciprocal**. A multiplicative inverse or reciprocal
of a real number *a* (except 0) is found by "flipping" *a *upside
down. The numerator of *a *becomes the
denominator of the reciprocal of *a* and the
denominator of *a* becomes the numerator of the
reciprocal of *a*.

**These two inverses will come in big time handy
when you go to solve equations later on. Keep them in your memory
bank until that time.**

** Example
1:** Write the opposite (or additive inverse) of -3.

When you take the reciprocal, the sign of the original number stays
intact. Remember that you need a number that when you multiply times
the given number you get 1. If you change the sign when you take
the reciprocal, you would get a -1, instead of 1, and that is a no no.

*a + b = b + a and
ab = ba*

The Commutative Property, in general, states that changing the ORDER
of two numbers either being added or multiplied, does NOT change
the value of it. The two sides are called equivalent expressions
because they look different but have the same value.

Using the commutative property of addition (where changing the order
of a sum does not change the value of it) we get

**2.5x + 3y = 3y + 2.5x.**

Using the communicative property of multiplication (where changing
the order of a product does not change the value of it), we get

**The Associative Properties of **

**Addition and Multiplication**

*a + (b + c) = (a + b) + c and
a(bc) = (ab)c*

The Associative property, in general, states that changing the GROUPING
of numbers that are either being added or multiplied does NOT change the
value of it. Again, the two sides are equivalent to each other.

**At this point it is good to remind you that
both the commutative and associative properties do NOT work for subtraction
or division.**

** Example
7:** Use the associative property to write an equivalent
expression to (a + 5b) + 2c.

Using the associative property of addition (where changing the grouping
of a sum does not change the value of it) we get

**(a + 5b) + 2c = a + (5b + 2c).**

Using the associative property of multiplication (where changing the
grouping of a product does not change the value of it) we get

**(1.5x)y = 1.5(xy)**

*a(b + c) = ab + ac *

or

*(b + c)a = ba + ca*

In other words, when you have a term being multiplied times two or
more terms that are being added (or subtracted) in a ( ), multiply the
outside term times EVERY term on the inside. Remember terms are separated
by + and -. This idea can be extended to more than two terms in the
( ).

** Example
9: **Use the distributive property to find the product - (5*x *+ 7).

Basically, when you have a negative sign in front of a ( ), like this
example, you can think of it as taking a -1 times the ( ). What you
end up doing in the end is taking the opposite of every term in the ( ).

As mentioned above, you can extend the distributive property to as
many terms as are inside the ( ). The basic idea is that you multiply
the outside term times EVERY term on the inside.

You can use any of these properties forwards or backwards, and that
includes the distributive property. When it says to simplify, that
means we want to write it in equivalent form that is more compact - get
it down to as few terms as possible.

Of course, we can combine the -10 and -12. But with the help of
the distributive property in reverse, we can also combine 12*x *and
-15*x*.

**Let's check it out:**

It makes sense that an equality involves an equal sign.

Here are some key words that translate into an = when writing out mathematical statements:

**Equals, is, represents, is the same as, **

**gives, yields, amounts to, is equal to.**

FYI, when you put an = between two mathematical expressions, you have
yourself an equation.

**Not Equal **

The above symbol is used when you want to say that two expressions
are not equal to each other.

**Read left to right:**

*a* < *b * *a* is less than *b*

*a* __<__ *b* *a* is less than or equal to *b*

*a* >* b* *a* is greater than *b*

*a* __>__*ba* is greater than or equal to *b *

If *a *is less than *b*,
that means *a* lies to the left of *b *on
the real number line.

If *a* is greater than *b*,
that means *a* lies to the right of *b* on the real number line.

The product of 5 and x is the same as 15.

As covered in tutorial 2, **product **translates into multiplication.

What will we use for **the same as**? If you said **=**,
you are correct!!

**Let's put everything together going left to right:**

**The product of 5 and x is the same as 15**

The sum of 3 and

Do you remember what **sum** translates into? If you
said **add**, you are doing great.

**Is less than** will need to be replaced by the symbol <.

**Let's put everything together going left to right:**

**The sum of 3 and y is less than 12.**

Twice the difference of 4 and

I**s less than or equal** to will need to be replaced by the symbol __<__**.**

The reciprocal of 5 is 1/5.

**Let's put everything together going left to right:**

**Twice the difference of 4 and a is less
than or equal to the reciprocal of 5.**

The quotient of

Do you remember what **quotient** translates into? I believe
that it is **division**, don't you agree?

**Is greater than **will need to be replaced by the symbol **>**.

What is the **opposite of 1**? Why, it is **-1**.

**Let's put everything together going left to right:**

**The quotient of x and 2 is greater than
the opposite of 1.**

3 times the sum of 2 and x is greater than or equal to 10.

**Is greater than or equal to** will need to be replaced by the symbol __>__

**Let's put everything together going left to right:**

**3 times the sum of 2 and x is greater than or equal to 10.**

The difference of x and 5 is not equal to 10.

**Is not equal to** will need to be replaced by the symbol

**Let's put everything together going left to right:**

**The difference of x and 5 is not equal to 10.**

The cost of

If each pizza costs 8.55, then, in order to find the cost, we would
have to multiply the number of pizzas (*x*) by the cost per pizza
(8.55).

**Hence, we would get the algebraic expression 8.55 x.**

Since -5 is to the left of 0 on the number line, then -5 is less than
0:

**-5 < 0**

Since -3.5 is to the right of - 4.5 on the number line, then
-3.5 is greater than - 4.5:

**-3.5 > - 4.5**

Since both 10/2 and 15/3 simplify to be 5, then 10/2 equals 15/3:

**10/2 = 15/3**

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1b: Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of each number.

1a. -7

(answer/discussion to 1a)

(answer/discussion to 1a)

1b. 3/5

(answer/discussion to 1b)

(answer/discussion to 1b)

Practice Problems 2a - 2b:Use a commutative property to write an equivalent expression.

Practice Problems 3a - 3b:Use an associative property to write an equivalent expression.

Practice Problems 4a - 4b:Use the distributive property to find the product.

Practice Problem 5a:Simplify the expression.

Practice Problems 6a - 6d:Write each statement using mathematical symbols.

6d. The quotient of x and 11 is less than - 4.

(answer/discussion to 6d)

(answer/discussion to 6d)

Practice Problem 7a:Write the following as an algebraic expression.

7a. Two angles are complimentary if the sum of their measures
is 90 degrees. If the measure of one angle is *x* degrees, represent the measure of the other angle as an expression of *x*.

(answer/discussion to 7a)

(answer/discussion to 7a)

Practice Problems 8a - 8c:Insert <, > or = to form a true statement.

8a. ½ ? .5

(answer/discussion to 8a)

(answer/discussion to 8a)

8b. -3.5 ? 1.5

(answer/discussion to 8b)

(answer/discussion to 8b)

8c. 4.3 ? 4.21

(answer/discussion to 8c)

(answer/discussion to 8c)

** Need Extra Help on these Topics?**

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm****#commutativeproperty**

This website helps with the commutative property.

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#associativeproperty**

This website helps with the associative property.

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#distributiveproperty**

This website helps with the distributive property.

**http://home.earthlink.net/~djbach/basic.html#anchor904011**

This website goes over the commutative, associative, and distributive
properties.**
**

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on June 11, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.