chapter 2 review algebra 1 algebraic expressions an algebraic expression is a collection of real...
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Chapter 2 Review
Algebra 1
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Algebraic Expressions
An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols.
Here are some examples of algebraic expressions.
27,7
5
3
1,4,75 2 xxyxx
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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.
75 2 xx
7,,5 2 xx
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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.
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Distributive Property
a ( b + c ) = ba + ca
To simplify some expressions we may need to use the Distributive Property
Do you remember it?
Distributive Property
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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 + 12
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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 + 14
Notice when a negative is distributed all the signs of the terms in the ( )’s change.
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Examples with 1 and –1.
Example 3: (x – 2)
= 1( x – 2 )
= x(1) – 2(1)
= x - 2
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.
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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.
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Examples
Like Terms
5x , -14x
-6.7xy , 02xy
The variable factors are
identical.
Unlike Terms
5x , 8y
The variable factors are
not identical.
22 8,3 xyyx
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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
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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
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Collecting Like Terms Example
31316
terms.likeCombine
31334124
terms.theReorder
33124134
2
22
22
yxx
yxxxx
xxxyx
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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
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Simplifying Example
431062
1 xx
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Simplifying Example
Distribute. 43106
2
1 xx
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Simplifying Example
Distribute. 43106
2
1 xx
12353
3432
110
2
16
xx
xx
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Simplifying Example
Distribute.
Combine like terms.
431062
1 xx
12353
3432
110
2
16
xx
xx
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Simplifying Example
Distribute.
Combine like terms.
431062
1 xx
12353
3432
110
2
16
xx
xx
76 x
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Evaluating Expressions
Remember to use correct order of operations.
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.
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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.
6 + 45 – 20
51 – 20
31
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Evaluating Example
1and2when34Evaluate 22 yxyxyx
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Evaluating Example
Substitute in the numbers.
1and2when34Evaluate 22 yxyxyx
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Evaluating Example
Substitute in the numbers.
1and2when34Evaluate 22 yxyxyx
22 131242
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Evaluating Example
Remember correct order of operations.
1and2when34Evaluate 22 yxyxyx
22 131242
Substitute in the numbers.
131244
384
15
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Common Mistakes
Incorrect Correct
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Your Turn
• Find the product
1. (-8)(3)
2. (20)(-65)
3. (-15)
• Simplify the variable expression
4. (-3)(-y)
5. 5(-a)(-a)(-a)
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Your Turn
• Evaluate the expression:
6. -8x when x = 6
7. 3x2 when x = -2
8. -4(|y – 12|) when y = 5
9. -2x2 + 3x – 7 when x = 4
10. 9r3 – (- 2r) when r = 2
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Your Turn Solutions
1. -24
2. -1300
3. -9
4. 3y
5. -5a3
6. -48
7. 12
8. -28
9. -27
10. 76
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Find the product.
a. (9)(–3) b.
c. (–3)3 d.
1(8) ( 6)
2
1( 2) ( 3)( 5)
2
-27(–4)(–6)
24
(–3)(–3)(–3)(9)(–3)
–27
1(–3)(–5)(–3)(–5)
15
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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.
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Extra Example 3Evaluate the expression when x = –7.a. 2(–x)(–x)
2 ( 7) ( 7)
2 7 7
14 7
98
OR simplify first:
2(–x)(–x)
2x2
2(-7)2
2(49)98
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Extra Example 3 (cont.)Evaluate the expression when x = –7.
b. 25
7x
25( 7)
7
25 7
7
5(2)
10
25
7x
25( 7)
7 2
357
10
OR use the associative property:
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CheckpointFind the product.1. (–2)(4.5)(–10) 2. (–4)(–x)2
3. Evaluate the expression when x = –3:(–1• x)(x)
90 –4x2
–9
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Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×
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Commutative Properties
• Changing the order of the numbers in addition or multiplication will not change the result.
• Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a.
• Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.
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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)
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Distributive Property
Multiplication distributes over addition.
acabcba
5323523
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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.
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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.
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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.
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Multiplicative Inverse Property
• For each real number a there exists a
unique real number such that their
product is 1.
1
a
11
a
a
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Let’s play “Name that property!”
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State the property or properties that justify the following.
3 + 2 = 2 + 3
Commutative Property
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State the property or properties that justify the following.
10(1/10) = 1
Multiplicative Inverse Property
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State the property or properties that justify the following.
3(x – 10) = 3x – 30
Distributive Property
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State the property or properties that justify the following.
3 + (4 + 5) = (3 + 4) +
5 Associative Property
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State the property or properties that justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property
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3 + 7 = 7 + 3
Commutative Commutative Property of AdditionProperty of Addition
2.2.
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8 + 0 = 8
Identity Property of Identity Property of AdditionAddition
3.3.
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6 • 4 = 4 • 6
Commutative Property Commutative Property of Multiplicationof Multiplication
5.5.
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5 • 1 = 5
Identity Property of Identity Property of MultiplicationMultiplication
11.11.
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51/7 + 0 = 51/7
Identity Property of Identity Property of AdditionAddition
25.25.
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a + (-a) = 0
Inverse Property of Inverse Property of AdditionAddition
40.40.