Download Aim #7 - Manhasset Public Schools

Homework #6 Solutions: pgs. 18-19 (1-20)
1) D
2) D
3) C
4) A
5) B
6) A
7) Commutative
8) Multiplicative Identity
9) Commutative
10) Additive Identity
12) Associative
13) Commutative 14) Distributive
15) Additive inverse 16) Distributive
18) -1
19) 0
20) 0
11) Associative
17) Multiplicative Inverse
Aim #7: How do we complete proofs using the properties of real numbers?
Homework: Handout
Do Now: Fill in the following flow diagram with the correct properties.
3(y+x) + 2x
(3y + 3x) + 2x
3(x+y) + 2x
2x + 3(x+y)
(2x + 3x) + 3y
2x + (3x + 3y)
Fill in the blanks of this proof showing that (w + 5)(w + 2) is equivalent to
2
w + 7w + 10. Write either commutative, associative, or distributive property in
each blank.
1) Show that 4y - 8(1 - y) = 12y - 8. Justify each step with a property.
4y - 8(1 - y) ______________________
4y - 8 + 8y _______________________
4y + 8y - 8 _______________________
(4y + 8y) - 8 ______________________
12y - 8
2) Draw a flow diagram and use it to write a proof that (xy)z = (zy)x for all real
numbers x, y, and z.
3) Justify each step with one of the properties of real numbers:
a) 12 + 3(a + 2b) + (-3a)
12 + 3a + 6b + (-3a)
12 + 3a + (-3a) + 6b
12 + 0 + 6b
12 + 6b
Given
__________________________
__________________________
__________________________
__________________________
b) 3x + 5 + (-5) + 2x
3x + 0 + 2x
3x + 2x
5x
Given
__________________________
__________________________
4) Using the Distributive Property, write an equivalent expression to 5(x - 6).
5) Using the Commutative Property, write an equivalent expression for 5(7x).
6) Does the Associative Property work over subtraction? Show an example to
support your answer.
7) What is the additive inverse of -8x? ___________
8) Justify each step with one of the properties of real numbers:
3a2(2a2 + 3) - 2(a4 + 8) Given
6a4 + 9a2 - 2a4 - 16
_____________________________
4
4
2
6a - 2a + 9a - 16
_____________________________
4a4 + 9a2 - 16
9) Justify each step with one of the properties of real numbers:
Given
2x2 + (5 + 3x)6 - 30
2x2 + 6(5 + 3x) - 30
___________________________
2x2 + 30 + 18x - 30
___________________________
2
___________________________
2x + 18x + 30 - 30
___________________________
2x2 + 18x + 0
2x2 + 18x
10) Write an equivalent expression for -4(5x + 9),
a. Using the Commutative Property of addition. __________________
b. Using the Commutative Property of multiplication. _________________,
11) Using the Associative Property of addition, write an equivalent expression for
3 + [(x + 5) + (x - 6)].
12) Using the Distributive Property, write two equivalent expressions for
4x3 - 8x2 + 4.
Sum It Up!
When writing proofs for solving equations, you must justify each step with a
property.