The Identity and Inverse Properties

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The Identity and Inverse Properties
Lesson 1.1.4
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Lesson 1.1.4
The Identity and Inverse Properties
California Standard Algebra and Functions
1.3 Simplify numerical expressions by applying
properties of rational numbers (e.g., identity,
inverse, distributive, associative, commutative)
and justify the process used.
What it means for you
Youll learn how to use math properties to show
why the steps of your work are reasonable.
Key Words

• justify
• identity
• inverse
• reciprocal
• multiplicative
• additive

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Lesson 1.1.4
The Identity and Inverse Properties
When youre simplifying and evaluating
expressions you need to be able to justify your
work.
To justify it means to use known math properties
to explain why each step of your calculation is
valid.
The math properties describe the ways that
numbers and variables in expressions behave you
need to know their names so that you can say
which one youre using for each step.
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Lesson 1.1.4
The Identity and Inverse Properties
The Identity Doesnt Change the Number
There are two identity properties one property
for addition and one property for multiplication
Adding 0 to a number doesnt change it. For
example 5 0 5 x 0 x
This is called the identity property of addition,
and 0 is called the additive identity.
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Lesson 1.1.4
The Identity and Inverse Properties
The second identity property is for
multiplication
Multiplying a number by 1 doesnt change it. For
example 1 7 7 1 x x
This is called the identity property of
multiplication, and 1 is called the
multiplicative identity.
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Lesson 1.1.4
The Identity and Inverse Properties
Guided Practice
1. What do you get by multiplying 6 by the
multiplicative identity? 2. What do you get by
adding the additive identity to 3y? Complete the
expressions in Exercises 36.
6
3y
3. x __ x 5. k __ k
4. __ 0 h 6. 1 __ t
0
h
1
t
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
The Inverse Changes the Number to the Identity
There are two inverse properties one for
addition and one for multiplication.
Different numbers have different additive
inverses and different multiplicative inverses.
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Lesson 1.1.4
The Identity and Inverse Properties
The Additive Inverse Adds to Give 0
The additive inverse is what you add to a number
to get 0.
The additive inverse of 2 is 2 2 2
0
The additive inverse of 3 is 3 3 3
0
The additive inverse of is 0
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Lesson 1.1.4
The Identity and Inverse Properties
Guided Practice
Give the additive inverses of the numbers in
Exercises 712.
7. 6 9. 5 11.
8. 19 10. 165 12.
6
19
5
165
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
The Multiplicative Inverse Multiplies to Give 1
The multiplicative inverse is what you multiply a
number by to get 1. So, a numbers
multiplicative inverse is one divided by the
number.
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Lesson 1.1.4
The Identity and Inverse Properties
The general rule for the multiplicative inverse
is
A multiplicative inverse is sometimes called a
reciprocal.
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Lesson 1.1.4
The Identity and Inverse Properties
Guided Practice
Give the multiplicative inverses of the numbers
in Exercises 1316.
13. 2 15. 4
14. 10 16. 5
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Fractions Have Multiplicative Inverses Too
When you multiply two fractions together, you
multiply their numerators and their denominators
separately.
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Lesson 1.1.4
The Identity and Inverse Properties
If you multiply a fraction by its multiplicative
inverse, the product will be 1 because thats
the definition of a multiplicative inverse.
For a fraction to equal 1, the numerator and
denominator must be the same.
So when a fraction is multiplied by its inverse,
the product of the numerators must be the same as
the product of the denominators.
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Lesson 1.1.4
The Identity and Inverse Properties
You can say that for any two non-zero numbers a
and b
The multiplicative inverse, or reciprocal, of a
fraction is just the fraction turned upside down.
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Lesson 1.1.4
The Identity and Inverse Properties
Example 1
Solution
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Guided Practice
Give the multiplicative inverses of the fractions
in Exercises 1720.
17. 19.
18. 20.
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
You Can Use Math Properties to Justify Your Work
To justify your work you need to use known math
properties to explain why each step of your
calculation is valid.
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Lesson 1.1.4
The Identity and Inverse Properties
Example 2
Solution
Write out the equation
Now justify each step
4 2 4 x
The distributive property
4 2 4 x
Get rid of the parentheses
The inverse property of multiplication
8 1x
8 1x
Do the multiplications
The identity property of multiplication
8 x
Do the subtraction
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Guided Practice
Simplify the expressions in Exercises 2124.
Justify your work.
21. m 1 6 22. d 0 d 9 23. (9 3f
) 24. 5(2 4 a) 10
m 6 using the multiplicative identity
d d 9 using the additive identity 9 using
the additive inverse
5(2 a) 10 using the multiplicative
inverse 10 5a 10 using the distributive
property 5a using the additive inverse
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Independent Practice
Complete the expressions in Exercises 14.
1. 1 __ 53. 2.5 __ 2.5
2. __ 0 24. 0.5 __ 0.5
5
2
1
0
Give the additive and multiplicative inverses of
the numbers in Exercises 58.
5. 6 7.
6. 7 8.
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Independent Practice
Simplify the expressions in Exercises 910.
Justify your work.
9. a a a 10. 4 d
a using the additive inverse
1d using the multiplicative inverse d using
the multiplicative identity
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Independent Practice
Simplify the expressions in Exercises 1112.
Justify your work.
11. 5( 2 n) 12. 2( x 4 0) 1(5
5 7)
Solution follows
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Lesson 1.1.4
The Identity and Inverse Properties
Round Up
The identity property and the inverse property
are two math properties youll need to use in
justifying your work.
Justifying your work is explaining how you know
that each step is right.
In the next Lesson youll cover two more
properties that can be used in justifying your
work.