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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.

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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.

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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**

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

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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

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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 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)**

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.

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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**

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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

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Simplifying Example Distribute.

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Simplifying Example Distribute.

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Simplifying Example Distribute. Combine like terms.

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Simplifying Example Distribute. Combine like terms.

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**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.

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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. – 20 51 – 20 31

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Evaluating Example

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Evaluating Example Substitute in the numbers.

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Evaluating Example Substitute in the numbers.

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**Evaluating Example Substitute in the numbers.**

Remember correct order of operations.

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Common Mistakes Incorrect Correct

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**Your Turn Find the product (-8)(3) (20)(-65) (-15)**

Simplify the variable expression (-3)(-y) 5(-a)(-a)(-a)

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**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

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

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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

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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 3 Evaluate the expression when x = –7. a. 2(–x)(–x)**

OR simplify first: 2(-7)2 2(49) 98

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**Extra Example 3 (cont.) Evaluate the expression when x = –7. b.**

OR use the associative property:

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**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

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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: = 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.

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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.

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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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**Commutative Property of Addition**

2. Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

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**Identity Property of Addition**

3. Which Property? 8 + 0 = 8 Identity Property of Addition

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**Commutative Property of Multiplication**

5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

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**Inverse Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Associative Property of Addition**

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

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

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**Distributive Property**

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

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**Associative Property of Multiplication**

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

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**Identity Property of Multiplication**

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

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Properties Using Negatives

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**Distributive Property**

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

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**Identity Property of Multiplication**

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

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**Inverse Property of Addition**

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

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**Associative Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Identity Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Addition**

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

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**Commutative Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Multiplication**

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

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**Commutative Property of Multiplication**

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

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Properties Using Fractions

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**Identity Property of Addition**

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

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**Commutative Property of Addition**

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

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**Identity Property of Multiplication**

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

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**(fraction)(fraction) = fraction**

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

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**Identity Property of Addition**

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

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**Associative Property of Multiplication**

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

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Properties Using Variables

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**Distributive Property**

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

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**Commutative Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Identity Property of Addition**

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

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**Associative Property of Multiplication**

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

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**Identity Property of Multiplication**

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

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**Commutative Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Addition**

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

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**Inverse Property of Addition**

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

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