1. Chapter 2 Review
Algebra 1

2. Algebraic Expressions
An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols.
Here are some examples of algebraic expressions.

3. Consider the example:
The terms of the expression are separated by addition. There are 3 terms in this example and they are
The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1.
The last term , -7, is called a constant since there is no variable in the term.

4. Let’s begin with a review of two important skills for simplifying expression, using the Distributive Property and combining like terms. Then we will use both skills in the same simplifying problem.

5. Distributive Property
To simplify some expressions we may need to use the Distributive Property
Do you remember it?
Distributive Property
a ( b + c ) = ba + ca

6. Examples
Example 1: 6(x + 2) Distribute the 6. 6 (x + 2) = x(6) + 2(6) = 6x + 12
Example 2: -4(x – 3)
Distribute the –4.
-4 (x – 3) = x(-4) –3(-4)
= -4x

7. Practice Problem Try the Distributive Property on -7 ( x – 2 ) .
Be sure to multiply each term by a –7.
-7 ( x – 2 ) = x(-7) – 2(-7)
= -7x
Notice when a negative is distributed all the signs of the terms in the ( )’s change.

8. Examples with 1 and –1. Example 3: (x – 2) = 1( x – 2 ) = x(1) – 2(1)
Notice multiplying by a 1 does nothing to the expression in the ( )’s.
Example 4: -(4x – 3)
= -1(4x – 3)
= 4x(-1) – 3(-1)
= -4x + 3
Notice that multiplying by a –1 changes the signs of each term in the ( )’s.

9. Like Terms
Like terms are terms with the same variables raised to the same power.
Hint: The idea is that the variable part of the terms must be identical for them to be like terms.

10. Examples Like Terms 5x , -14x -6.7xy , 02xy The variable factors are
identical.
Unlike Terms
5x , 8y
The variable factors are
not identical.

11. Combining Like Terms Recall the Distributive Property
a (b + c) = b(a) +c(a)
To see how like terms are combined use the
Distributive Property in reverse.
5x + 7x = x (5 + 7)
= x (12)
= 12x

12. Example All that work is not necessary every time.
Simply identify the like terms and add their
coefficients.
4x + 7y – x + 5y = 4x – x + 7y +5y
= 3x + 12y

13. Collecting Like Terms Example

14. Both Skills This example requires both the Distributive
Property and combining like terms.
5(x – 2) –3(2x – 7)
Distribute the 5 and the –3.
x(5) - 2(5) + 2x(-3) - 7(-3)
5x – 10 – 6x + 21
Combine like terms.
- x+11

15. Simplifying Example

16. Simplifying Example
Distribute.

17. Simplifying Example
Distribute.

18. Simplifying Example
Distribute.
Combine like terms.

19. Simplifying Example
Distribute.
Combine like terms.

20. Evaluating Expressions
Evaluate the expression 2x – 3xy +4y when
x = 3 and y = -5.
To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.
Remember to use correct order of operations.

21. Example Evaluate 2x–3xy +4y when x = 3 and y = -5.
Substitute in the numbers.
2(3) – 3(3)(-5) + 4(-5)
Use correct order of operations.
– 20
51 – 20 31

22. Evaluating Example

23. Evaluating Example
Substitute in the numbers.

24. Evaluating Example
Substitute in the numbers.

25. Evaluating Example Substitute in the numbers.
Remember correct order of operations.

26. Common Mistakes
Incorrect
Correct

27. Your Turn Find the product (-8)(3) (20)(-65) (-15)
Simplify the variable expression
(-3)(-y)
5(-a)(-a)(-a)

28. Your Turn Evaluate the expression: -8x when x = 6 3x2 when x = -2
-4(|y – 12|) when y = 5
-2x2 + 3x – 7 when x = 4
9r3 – (- 2r) when r = 2

29. Your Turn Solutions
-24
-1300 -9 3y
-5a3
-48 12
-28
-27 76

30. Find the product.
a. (9)(–3) b.
c. (–3) d.
-27
(–4)(–6) 24
(–3)(–3)(–3)
1(–3)(–5)
(9)(–3)
(–3)(–5)
–27 15

31. Find the product.
a. (–n)(–n)
b. (–4)(–x)(–x)(x)
c. –(b)3
d. (–y)4
Two negative signs: n2
Three negative signs:
–4x3
One negative sign:
–(b)(b)(b) = –b3
Four negative signs:
(–y)(–y)(–y)(–y) = y4
SUMMARY:
An even number of negative signs results in a
positive product, and an odd number of negative signs results in a negative product.

32. Extra Example 3 Evaluate the expression when x = –7. a. 2(–x)(–x)
OR simplify first:
2(-7)2
2(49) 98

33. Extra Example 3 (cont.) Evaluate the expression when x = –7. b.
OR use the associative property:

34. Checkpoint Find the product. 1. (–2)(4.5)(–10) 2. (–4)(–x)2
3. Evaluate the expression when x = –3:
(–1• x)(x) 90
–4x2 –9

35. Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×

36. Commutative Properties
Changing the order of the numbers in addition or multiplication will not change the result.
Commutative Property of Addition states: = or a + b = b + a.
Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

37. Associative Properties
Changing the grouping of the numbers in addition or multiplication will not change the result.
Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c
Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

38. Distributive Property
Multiplication distributes over addition.

39. Additive Identity Property
There exists a unique number 0 such that zero preserves identities under addition.
a + 0 = a and 0 + a = a
In other words adding zero to a number does not change its value.

40. Multiplicative Identity Property
There exists a unique number 1 such that the number 1 preserves identities under multiplication.
a ∙ 1 = a and 1 ∙ a = a
In other words multiplying a number by 1 does not change the value of the number.

41. Additive Inverse Property
For each real number a there exists a unique real number –a such that their sum is zero.
a + (-a) = 0
In other words opposites add to zero.

42. Multiplicative Inverse Property
For each real number a there exists a unique real number such that their product is 1.

43. Let’s play “Name that property!”

44. State the property or properties that justify the following.
3 + 2 = 2 + 3
Commutative Property

45. State the property or properties that justify the following.
10(1/10) = 1
Multiplicative Inverse Property

46. State the property or properties that justify the following.
3(x – 10) = 3x – 30
Distributive Property

47. State the property or properties that justify the following.
3 + (4 + 5) = (3 + 4) + 5
Associative Property

48. State the property or properties that justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property

49. Commutative Property of Addition2.
Which Property?
3 + 7 = 7 + 3
Commutative Property of Addition

50. Identity Property of Addition3.
Which Property?
8 + 0 = 8
Identity Property of Addition

51. Commutative Property of Multiplication5.
Which Property?
6 • 4 = 4 • 6
Commutative Property of Multiplication

52. Inverse Property of Addition6.
Which Property?
17 + (-17) = 0
Inverse Property of Addition

53. Commutative Property of Multiplication7.
Which Property?
2(5) = 5(2)
Commutative Property of Multiplication

54. Associative Property of Addition1.
Which Property?
(2 + 1) + 4 = 2 + (1 + 4)
Associative Property of Addition

55. 8.
Which Property?
even + even = even
Closure Property

56. Distributive Property9.
Which Property?
3(2 + 5) = 3•2 + 3•5
Distributive Property

57. Associative Property of Multiplication
10.
Which Property?
6(7•8) = (6•7)8
Associative Property of Multiplication

58. Identity Property of Multiplication
11.
Which Property?
5 • 1 = 5
Identity Property of Multiplication

59. Properties Using
Negatives

60. Distributive Property
13.
Which Property?
(6 – 3)4 = 6•4 – 3•4
Distributive Property

61. Identity Property of Multiplication
14.
Which Property?
1(-9) = -9
Identity Property of Multiplication

62. Inverse Property of Addition
15.
Which Property?
3 + (-3) = 0
Inverse Property of Addition

63. Associative Property of Addition
16.
Which Property?
1 + [-9 + 3]
= [1 + (-9)] + 3
Associative Property of Addition

64. Commutative Property of Multiplication
17.
Which Property?
-3(6) = 6(-3)
Commutative Property of Multiplication

65. Identity Property of Addition
18.
Which Property?
= -8
Identity Property of Addition

66. Distributive Property
19.
Which Property?
3•7 – 3•4 = 3(7 – 4)
Distributive Property

67. Associative Property of Addition
20.
Which Property?
6 + [(3 + (-2)]
= (6 + 3) + (- 2)
Associative Property of Addition

68. Commutative Property of Addition
21.
Which Property?
7 + (-5) =
Commutative Property of Addition

69. Distributive Property
22.
Which Property?
(5 + 4)9 =
Distributive Property

70. Associative Property of Multiplication
23.
Which Property?
-3(5 • 4) = (-3 • 5)4
Associative Property of Multiplication

71. Commutative Property of Multiplication
24.
Which Property?
-8(4) = 4(-8)
Commutative Property of Multiplication

72. Properties Using
Fractions

73. Identity Property of Addition
25.
Which Property?
51/7 + 0 = 51/7
Identity Property of Addition

74. Commutative Property of Addition
26.
Which Property?
3/4 – 6/7 = – 6/7 + 3/4
Commutative Property of Addition

75. Identity Property of Multiplication
27.
Which Property?
12/5 • 1 = 12/5
Identity Property of Multiplication

76. (fraction)(fraction) = fraction
28.
Which Property?
(fraction)(fraction)
= fraction
Closure Property

77. Identity Property of Addition
29.
Which Property?
-8 2/5 + 0 = -8 2/5
Identity Property of Addition

78. Associative Property of Multiplication
30.
Which Property?
[(-2/3)(5)]9 = -2/3[(5)(9)]
Associative Property of Multiplication

79. Properties Using
Variables

80. Distributive Property
31.
Which Property?
6(3 – 2n) = 18 – 12n
Distributive Property

81. Commutative Property of Addition
32.
Which Property?
2x + 3 = 3 + 2x
Commutative Property of Addition

82. Commutative Property of Multiplication
33.
Which Property?
ab = ba
Commutative Property of Multiplication

83. Identity Property of Addition
34.
Which Property?
a + 0 = a
Identity Property of Addition

84. Associative Property of Multiplication
35.
Which Property?
a(bc) = (ab)c
Associative Property of Multiplication

85. Identity Property of Multiplication
36.
Which Property?
a•1 = a
Identity Property of Multiplication

86. Commutative Property of Addition
37.
Which Property?
a +b = b + a
Commutative Property of Addition

87. Distributive Property
38.
Which Property?
a(b + c) = ab + ac
Distributive Property

88. Associative Property of Addition
39.
Which Property?
a + (b + c) = (a +b) + c
Associative Property of Addition

89. Inverse Property of Addition
40.
Which Property?
a + (-a) = 0
Inverse Property of Addition