Learning Objectives
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
Multiplication identity is 1
a(1) = 1(a) = a
For each real number a, there is a unique real number,
denoted -a,
such that
a + (-a) = 0.
For each real number a, except 0, there is a unique real number such that
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
2.5x + 3y = 3y + 2.5x.
The Associative Properties of
Addition and Multiplication
a + (b + c) = (a + b) + c and
a(bc) = (ab)c
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.
(a + 5b) + 2c = a + (5b + 2c).
(1.5x)y = 1.5(xy)
a(b + c) = ab + ac
or
(b + c)a = ba + ca
Example
9: Use the distributive property to find the product - (5x + 7).
Of course, we can combine the -10 and -12. But with the help of the distributive property in reverse, we can also combine 12x and -15x.
Let's check it out:
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
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 greater than b,
that means a lies to the right of b on the real number line.
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
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.
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.
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.
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.
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.
Hence, we would get the algebraic expression 8.55x.
-5 < 0
-3.5 > - 4.5
10/2 = 15/3
Practice Problems
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.
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.
Practice Problem 7a: Write the following as an algebraic expression.
Practice Problems 8a - 8c: Insert <, > or = to form a true statement.
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.