1. The Identity and Inverse Properties
Lesson 1.1.4

2. The Identity and Inverse Properties
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:
You’ll learn how to use math properties to show why the steps of your work are reasonable.
Key Words:
justify
identity
inverse
reciprocal
multiplicative
additive

3. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
When you’re 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 you’re using for each step.

4. The Additive Identity = 0
Lesson
1.1.4
The Identity and Inverse Properties
The Identity Doesn’t Change the Number
There are two identity properties — one property for addition and one property for multiplication:
The Additive Identity = 0
For any number, a, a + 0 = a.
Adding 0 to a number doesn’t change it.
For example: = 5 x + 0 = x
This is called the identity property of addition, and 0 is called the additive identity.

5. The Multiplicative Identity = 1
Lesson
1.1.4
The Identity and Inverse Properties
The second identity property is for multiplication:
The Multiplicative Identity = 1
For any number, a, a • 1 = a.
Multiplying a number by 1 doesn’t 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.

6. The Identity and Inverse Properties
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 3–6. 6 3y
3. x + __ = x
5. k • __ = k
4. __ + 0 = h
6. 1 • __ = t h 1 t
Solution follows…

7. The Identity and Inverse Properties
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.

8. The Additive Inverse of a is –a.
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 = 0
The additive inverse of –3 is –3 + 3 = 0 1 4 1 4 1 4 1 4
The additive inverse of is – – = 0
The Additive Inverse of a is –a.
For any number, a, a + –a = 0.

9. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Guided Practice
Give the additive inverses of the numbers in Exercises 7–12.
7. 6
9. –5
11.
8. 19
10. –165
12. – –6
–19 5
165 2 3 – 1 7 2 3
Solution follows…

10. The Identity and Inverse Properties
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 number’s multiplicative inverse is one divided by the number.
The multiplicative inverse of 2 is 1 ÷ 2 = . 1 2 1 7
The multiplicative inverse of 7 is 1 ÷ 7 = . 2 1 7
2 • 1
7 • 1
To check, 2 • = = = 1 and 7 • = = = 1.

11. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
The general rule for the multiplicative inverse is:
The Multiplicative Inverse of a is .
For any nonzero number, a, a • = 1. 1 a
A multiplicative inverse is sometimes called a reciprocal.

12. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Guided Practice
Give the multiplicative inverses of the numbers in Exercises 13–16.
13. 2
15. –4 1 2
14. 10
16. –5 1 10 – 1 4 – 1 5
Solution follows…

13. The Identity and Inverse Properties
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.
For example: • = = 3 4 8 1 2
1 • 3
2 • 4

14. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
If you multiply a fraction by its multiplicative inverse, the product will be 1 — because that’s 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.
For example: • = = = 1 4 3 12
3 • 4
4 • 3

15. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
You can say that for any two non-zero numbers a and b: a b
• = = 1 ab
So, the multiplicative inverse of is . a b
The multiplicative inverse, or reciprocal, of a fraction is just the fraction turned upside down.

16. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Example 1
Give the multiplicative inverse of . 1 4
Solution
, or 4, is the multiplicative inverse of . 1 4
To check your answer multiply it by . 1 4
• = = = 1 4 1
1 • 4
4 • 1
So 4 is the multiplicative inverse of . 1 4
Solution follows…

17. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Guided Practice
Give the multiplicative inverses of the fractions in Exercises 17–20.
17.
19. –
18.
20. – 2 5 1 10 5 2 10 1
or 10 3 4 1 2 – 4 3 – 2 1
or –2
Solution follows…

18. The Identity and Inverse Properties
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.

19. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Example 2
Simplify the expression 4(2 – x). Justify your work. 1 4
Solution
4(2 – x) 1 4
Write out the equation
Now justify each step 1 4
= 4 • 2 – 4 • x
= 4 • 2 – 4 • x
Get rid of the parentheses
The distributive property
= 8 – 1x
= 8 – 1x
The inverse property of multiplication
Do the multiplications
= 8 – x
Do the subtraction
The identity property of multiplication
Solution follows…

20. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Guided Practice
Simplify the expressions in Exercises 21–24. Justify your work.
21. m • 1 + 6
22. d + 0 – d + 9
(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 1 3 1 3
= • • 3f using the distributive property
= 3 + f using the multiplicative inverse 1 4
= –5(2 – a) + 10 using the multiplicative inverse
= –10 + 5a + 10 using the distributive property
= 5a using the additive inverse
Solution follows…

21. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Independent Practice
Complete the expressions in Exercises 1–4.
1. 1 • __ = • __ = 2.5
2. __ + 0 = –2 4. –0.5 + __ = –0.5 5 –2 1
Give the additive and multiplicative inverses of the numbers in Exercises 5–8.
5. 6 7.
6. –7
8. – 1 6
–6, 1 7
7, – 5 7 2 3 5 7
– , 2 3
, –
Solution follows…

22. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Independent Practice
Simplify the expressions in Exercises 9–10. Justify your work.
9. a + a – a
• 4 • d
= a using the additive inverse 1 4
= 1d using the multiplicative inverse
= d using the multiplicative identity
Solution follows…

23. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Independent Practice
Simplify the expressions in Exercises 11–12. Justify your work.
11. 5( – n)
12. 2( x ) – 1(5 – 5 + 7) 1 5
= 5 • • 2 – 5 • n using the distributive property
= – 5n using the multiplicative inverse
= 11 – 5n 1 5 1 2 1 2
= 2 • x + 2 • • 0 – 1(5 – 5 + 7) using the distributive property
= 1x – 1(5 – 5 + 7) using the multiplicative inverse
= x – (5 – 5 + 7) using the multiplicative identity
= x + 8 – (5 – 5 + 7) using the additive identity
= x + 8 – 7 using the additive inverse
= x + 1
Solution follows…

24. The Identity and Inverse Properties
Lesson
1.1.4
The Identity and Inverse Properties
Round Up
The identity property and the inverse property are two math properties you’ll need to use in justifying your work.
Justifying your work is explaining how you know that each step is right.
In the next Lesson you’ll cover two more properties that can be used in justifying your work.