Are You Ready? Write each verbal expression as an algebraic expression more than m 2. r increased by 5

Transcription

1 Are You Ready? Write each verbal expression as an algebraic expression more than m 2. r increased by minus q 4. the difference of 20 and t 5. the sum of v and 8 6. the product of 4 and w Write each algebraic expression as a verbal expression n 8. z x d x 12. u + 8

2 Lesson 1A Algebraic Expressions To evaluate an algebraic expression you replace each variable with its numerical value, then use the order of operations to simplify. Example 1 Evaluate 6x 7 if x = 8. 6x 7 = 6(8) 7 Replace x with 8. = 48 7 Use the order of operations. = 41 Subtract 7 from 48. Example 2 Evaluate 5m 3n if m = 6 and n = 5. 5m 3n = 5(6) 3(5) Replace m with 6 and n with 5. = Use the order of operations. = 15 Subtract 15 from 30. Example 3 Evaluate ab ab = (7)(6) 3 3 = if a = 7 and b = 6. Replace a with 7 and b with 6. The fraction bar is like a grouping symbol. = 14 Divide. Example 4 Evaluate x if x = 3. x = Replace x with 3. = Use the order of operations. = 31 Add 27 and 4. Exercises Evaluate each expression if a = 4, b = 2, and c = ac 2. 5b 3 3. abc c 5. ab 8 6. b c a bc 9. 7c

3 Lesson 1B Algebraic Expressions Evaluate each expression if r = 5, s = 2, t = 7, and u = s u 3. u + r 4. 11t u 6. 4r 10s 7. 3u t r s r 12. (3+u)2 8 Evaluate each expression if a = 4.1, b = 5.7, and c = a + b c (a + b) 15. b c MOON The expression w gives the weight of an object on the Moon in pounds with a weight of w pounds 6 on Earth. What is the weight of a space suit on the Moon if the space suit weighs pounds on Earth? 17. Complete the table. Ounces Pounds (p) (16p)

4 Lesson 2A Sequences An arithmetic sequence is a list in which each term is found by adding the same number to the previous term. 1, 3, 5, 7, 9, Example 1 Describe the relationship between terms in the arithmetic sequence 17, 23, 29, 35, Then write the next three terms in the sequence. 17, 23, 29, 35,. Each term is found by adding 6 to the previous term. The next three terms are 41, 47, and = = = 53 Example 2 MONEY Brian s parents have decided to start giving him a monthly allowance for one year. Each month they will increase his allowance by $10. Suppose this pattern continues. What algebraic expression can be used to find Brian s allowance after any given number of months? How much money will Brian receive for allowance for the 10th month? Make a table to display the sequence. Position Operation Value of Term n n 10 10n Each term is 20 times its position number. So, the expression is 10n. How much money will Brian receive after 10 months? 10n Write the expression. 10(10) = 100 Replace n with 10 So, Brian will receive $100 after 10 months. Exercises Describe the relationship between terms in the arithmetic sequences. Write the next three terms in the sequence. 1. 2, 4, 6, 8, 2. 4, 7, 10, 13, , 0.6, 0.9, 1.2, 4. SALES Mama s bakery just opened and is currently selling only two types of pastry. Each month, Mama s bakery will add two more types of pastry to their menu. Suppose this pattern continues. What algebraic expression can be used to find the number of pastries offered after any given number of months? How many pastries will be offered in one year?

5 Lesson 2B Sequences Describe the relationship between the terms in each arithmetic sequence. Then write the next three terms in each sequence. 1. 0, 5, 10, 15, 2. 1, 3, 5, 7, 3. 18, 27, 36, 45, , 0.8, 1.2, 1.6, , 3.7, 3.7, 3.7, , 6.2, 7.3, 8.4, 7. 17, 31, 45, 59, 8. 30, 50, 70, 90, 9. 14, 41, 68, 95, NUMBER SENSE Find the 40th term in each arithmetic sequence , 8, 12, 16, , 26, 39, 52, 12. 6, 12, 18, 24, 13. GEOMETRY The lengths of the sides of a 6-sided polygon are an arithmetic sequence. The length of the shortest side is 3 meters. If the length of the next longer side is 5 meters, what is the length of the longest side? 14. FREE FALLING OBJECT A free falling object increases speed by a little over 22 miles per hour each second. The arithmetic sequence 22, 44, 66,, represents the speed after each second, in miles per hour, of a dropped object. How fast is a rock falling after 8 seconds if it is dropped over the side of a cliff?

6 Lesson 3A Properties of Operations Example 1 Name the property shown by the statement u + v = v + u. The order in which the variables are being added changed. This is the Commutative Property of Addition. Example 2 State whether the following conjecture is true or false. If false, provide a counterexample. Subtraction of integers is commutative. Write two subtraction expressions using the Commutative Property State the conjecture. 8 8 Subtract. We found a counterexample. That is, So, subtraction is not commutative. The conjecture is false. Example 3 Simplify the expression. Justify each step. 9 + (3x + 4) 9 + (3x + 4) = 9 + (4 + 3x) Commutative Property of Addition = (9 + 4) + 3x Associative Property of Addition = x Simplify. Exercises Name the property shown by each statement = (3y + 2) = (4 + 3y) + 2 State whether the following conjectures are true or false. If false, provide a counterexample. 3. The product of two even numbers is odd. 4. The difference of two odd numbers is even. Simplify (5x + 2). Justify each step.

7 Name the property shown by each statement. Lesson 3B Properties of Operations 1. 1 (a + 3) = a p + (3q + 2) = (2p + 3q) (ab) c = c (ab) 4. 2t 0 = 0 5. m (nr) = (mn) r s = 2s State whether the following conjectures are true or false. If false, provide a counterexample. 7. The product of an odd number and an even number is always odd. 8. The sum of two whole numbers is always larger than either whole number. Simplify each expression. Justify each step. 9. 2d (3) 10. 2y + (4 + 5y) 11. FAXES Marcellus sent four faxes to Gem. The first fax took 14 seconds to send, the second fax 19 seconds, the third 16 seconds, and the fourth 11 seconds. Use mental math to find out how many seconds it took to fax all four documents to Gem. Explain your reasoning.

8 Practice Quiz 1 Evaluate each expression if d = -3, e = 2, f = 4 and g = 5. Show all work!! 1. 2(d + 9) 2. e f 4 3. d d (5+g)² g 2 Matching. Write the letter on the line of the matching algebraic expression for each phrase. In each case, let the variable x represent the unknown quantity feet less than the height A. x Sarah worked 5 more hours than Pam B. x Kumar has triple the number of goals as Jacob C. x Allison is 5 years younger than Nathan D. 3x 11. The quotient of a number and 3 E. x Triple the sum of a number and 5 F. 3(x + 5).

9 Describe the relationship between the terms in each arithmetic sequence. Then write the next three terms in each sequence , 18, 27, 36, , 8.5, 12.5, 16.5,. Relationship: Relationship:,,,, 15. Which expression can be used to find the nth term in this sequence? Month n Plant Height (in) ? A. n 3 B. n 3 C. 3n D. n + 3 Matching. Write the letter on the line of the matching property illustrated by each expression. You may use a property more than once. 16. a + (b + 2) = (b + 2) + a A. Commutative Property 17. f 0 = 0 B. Associative Property 18. (7 + x) + 0 = 7 + x C. Additive Identity Property (c + 1) = (6 + c) + 1 D. Multiplicative Identity Property 20. 4(ab) = (4a)b E. Multiplication Property of Zero 21. y 9 = 9y m = m

10 Lesson 4A The Distributive Property Words Distributive Property To multiply a sum or difference by number, multiply each term inside the parentheses by the number outside the parentheses. Symbols a (b + c) = ab + ac a (b c) = ab ac Examples 3( 2 + 5) = (8 3) = Examples Use the Distributive Property to evaluate each expression. 1 5(x + 3) 5 (x + 3) = 5 x Expand using the Distributive Property 2 (4x y)9 = 5x + 15 Simplify. (4x y) 9 = [4x + ( y)]9 Rewrite 4x y as 4x + ( y). = (4x)9 + ( y)9 Expand using the Distributive Property. = 36x + ( 9y) Simplify. = 36x 9y Definition of subtraction. Example 3 MOVIES Alwyn is taking three of his friends to the movies. Tickets cost $8.90 per person. Find Alwyn s total cost. You can use the Distributive Property to find the total cost mentally 4 ($9 $0.10) = 4 ($9) 4 ($0.10) Distributive Property = $36 $0.40 Multiply. = $35.60 Subtract. Alwyn will pay $35.60 for himself and three friends to go to the movies. Exercises Use the Distributive Property to evaluate or rewrite each expression. 1. 5(w + 4) 2. (x 5)( 2) 3. 7 (6x 2y) 4. 6(4 + 2m) 5. 8(2n + 7) 6. (3v + 6w)2 7. BOOKS Mariah bought 7 books costing $11.20 each. Find the total cost of the 7 books. Justify your answer by using the Distributive Property.

11 Lesson 4B The Distributive Property Use the Distributive Property to evaluate each expression. 1. (16 6) (12 + 3) 3. 3 ( 7 + 2) 4. (8 + 3) ( 1) Use the Distributive Property to rewrite each expression. 5. (2 + g) (h 5g) 7. 7 (5 n) 8. 8 (2m + 1) 9. 6x (y z) (2b 2a) 11. DINING OUT The table shows the different prices at a diner. Item Cost ($) a. Write two equivalent expressions for the total cost if Sandwich $5 two customers order each of the items. Drink $2 Dessert $3 b. What is the total cost for both customers? 12. SUNDAES Carmine bought 5 ice cream sundaes for his friends. If each sundae costs $4.95, how much did he spend? Justify your answer by using the Distributive Property.

12 Mid Chapter Review Define and give examples for the following vocabulary terms. 1. Variable 2. Algebraic Expression 3. Sequence 4. Term Match each property with their example by drawing a line to connect them. a + (b + 6) = (b + 6) + a Associative Property z 0 = 0 Additive Identity Property (3 + x) + 0 = 3 + x Multiplicative Identity Property 5 + (p + 1) = (5 + p) + 1 Multiplication Property of Zero 2(ab) = (2a)b Commutative Property y 8 = 8y 1 f = f

13 Lesson 5A Simplify Algebraic Expressions When a plus or minus sign separates an algebraic expression into parts, each part is called a term. The numerical factor of a term that contains a variable is called the coefficient of the variable. A term without a variable is called a constant. Like terms contain the same variables to the same powers, such as 3x 2 and 2x 2. Example 1 Identify the terms, like terms, coefficients, and constants in the expression 7x 5 + x 3x. 7x 5 + x 3x = 7x + ( 5) + x + ( 3x) Definition of subtraction = 7x + ( 5) + 1x + ( 3x) Identity Property; x = 1x The terms are 7x, 5, x, and 3x. The like terms are 7x, x, and 3x. The coefficients are 7, 1, and 3. The constant is 5. An algebraic expression is in simplest form if it has no like terms and no parentheses. Examples Write each expression in simplest form. 2 5x + 3x 5x + 3x = (5 + 3) x or 8x Distributive Property; simplify. 3 2m m 3 2m and 6m are like terms. 5 and 3 are also like terms. 2m m 3 = 2m m + ( 3) Definition of subtraction = 2m + 6m ( 3) Commutative Property = ( 2 + 6) m ( 3) Distributive Property = 4m + 2 Simplify. Exercises Identify the terms, like terms, coefficients, and constants in each expression. 1. 4y 3 + 2y 2. 5g g g a 4 a terms: terms: terms: like terms: like terms: like terms: coefficients: coefficients: coefficients: constants: constants: constants: Write each expression in simplest form. 4. 3d + 6d s z + 3 9z 8

14 Lesson 5B Simplify Algebraic Expressions Identify the terms, like terms, coefficients, and constants in each expression. 1. 4b + 7b t 3t + t 3. 5x + 4 x 1 terms: terms: terms: like terms: like terms: like terms: coefficients: coefficients: coefficients: constants: constants: constants: 4. 2z z h 8 h 6. y y terms: terms: terms: like terms: like terms: like terms: coefficients: coefficients: coefficients: constants: constants: constants: Write each expression in simplest form. 7. h + 6h 8. 10k k 9. 3b b x x c 3d 12c + d 12. y + 9z 16y 25z MEASUREMENT Write an expression in simplest form for the perimeter of each figure SHOPPING Maggie bought c CDs for $12 each, b books for $7 each, and a purse costing $24. a. Write an expression to show the total amount of money Maggie spent. b. If Maggie bought 4 CDs and 3 books, how much money did she spend?

15 Lesson 6A Factor Linear Expressions A linear expression is in factored form when it is expressed as the product of its factors. Example 1 Factor 5x Use the GCF to factor the linear expression. 5x = 5 x Write the prime factorization of 5x and = 5 2 Circle the common factors. The GCF of 5x and 10 is 5. Write each term as a product of the GCF and its remaining factors. 5x + 10 = 5(x) + 5(2) = 5(x + 2) Distributive Property So, 5x + 10 = 5(x + 2). Example 2 Factor 3x x = 3 x 8 = There are no common factors, so 3x + 8 cannot be factored. Exercises Factor each expression. If the expression cannot be factored, write cannot be factored x x x x x x x x + 13

16 Find the GCF of each pair of monomials. Lesson 6B Factor Linear Expressions 1. 20, 45x 2. 15r, xy, 14x 4. 16mn, 24m 5. 25f, 60g 6. 50j, 75jk Factor each expression. If the expression cannot be factored, write cannot be factored. 7. 4x r x x x x GEOMETRY The rectangle shown below has a total area of (4x + 36) square feet. Factor 4x x 36

17 Lesson 6a Factoring Linear Expressions Factor each term. Write the prime factorization. If the expression cannot be factored, write cannot be factored. Use the cake method x 8. 8b 9. 4x x n Find the GCF of each pair of monomials n and b and x 2 y and 54

18 Lesson 7A Add Linear Expressions You can use models to add linear expressions. Example 1 Add (3x + 5) + (2x + 3). Step 1 Model each expression. Example 2 Add (x 2) + ( 2x + 4). Step 1 Model each expression. Step 2 Combine like tiles and write an expression for the combined tiles. Step 2 Combine like tiles and write an expression for the combined tiles. So, (3x + 5) + (2x + 3) = 5x + 8. Step 3 Remove all zero pairs and write an expression for the remaining tiles. So, (x 2) + ( 2x + 4) = x + 2. Exercises Add. Use models if needed. 1. (5x + 2) + (3x + 1) 2. ( 8x + 1) + ( 2x + 6) 3. ( 7x + 4) + (x 5) 4. ( 6x + 1) + (4x 1)

19 Add. Use models if needed. Lesson 7B Add Linear Expressions 1. (9x + 7) + (x + 3) 2. ( 4x + 6) + (x 5) 3. ( 3x + 15) + ( 3x + 2) 4. ( 2x + 10) + ( 8x 1) 5. ( 2x + 4) + (x 11) 6. (8x + 9) + ( 6x 1) 7. (1.3x + 2.4) + ( 6.1x 3.2) 8. (0.5x 0.6) + (0.75x 0.1) 9. GEOMETRY A rectangle has side lengths of (3x + 6) inches and (2x 4) inches. Write an expression to represent the perimeter of the rectangle. 10. CRUISE SHIPS The table shows the number of cruise ships in a harbor on various days. Day Monday Tuesday Wednesday Thursday Friday Number x 4 x + 9 2x 3x 7 4 a. Write an expression for the total number of cruise ships in the harbor on Monday and Tuesday. b. Write an expression for the total number of cruise ships in the harbor on all 5 days.

20 Lesson 8A Subtract Linear Expressions When subtracting expressions, subtract like terms. You can use models or the additive inverse. Example 1 Find ( 3x 2) (4x). Step 1 Model the expression 3x 2. Step 2 Since there are no positive x-tiles to remove, add four zero pairs of x-tiles. Remove four positive x-tiles. So, ( 3x 2) (4x) = 7x 2. Example 2 Subtract (4x + 6) ( 7x + 1). The additive inverse of 7x + 1 is 7x 1. 4x + 6 Arrange like terms in columns. + 7x 1 Add. 11x + 5 So, (4x + 6) ( 7x + 1) = 11x + 5. Exercises Subtract. Use models if needed. 1. (9x + 10) (2x + 4) 2. (3x + 4) (2x 5) 3. (6x + 3) ( x 2) 4. (4x 1) (x + 3) 5. (3x 1) (2x 6)

21 Subtract. Use models if needed. Lesson 8B Subtract Linear Expressions 1. (9x + 7) (x + 3) 2. (3x 4) (x 5) 3. ( 3x + 15) ( 3x + 2) 4. ( 2x + 10) ( 8x 1) 5. ( 2x + 4) (x 11) 6. (8x + 9) (6x 1) 7. (x + 3) (5x 4) 8. (2x 4) ( x + 9) 9. FOOTBALL The Dolphins scored 2x 7 points, while the Jaguars scored 5x 3 points.how many more points did the Dolphins score than the Jaguars? 10. LUNCH The table shows the cost of a sandwich and a drink at a local cafeteria. How much more does a sandwich cost than a drink? Item Sandwich Drink Cost ($) 2x x COLLEGE COSTS The table shows some college costs. How much more is tuition than the cost of fees and room and board? Item Tuition Fees Room and Board Cost ($) 8x + 75 x + 50 x + 3

22 Lesson 8a Distributing a Negative Use the Distributive Property to evaluate each expression. 1. -(18 6) 2. -3(11 + 3) 3. ( 7 + 4) 4. (9 + 13) ( 1) Use the Distributive Property to rewrite each expression. 5. (6 + g) (-1) 6. -1(h 7g) 7. (4 n) (5m + 2) 9. -x(y z) (2b 2a)

23 Practice Quiz 2 Use the Distributive Property to evaluate or rewrite each expression in simplest form. Show all work!! 1. 15(10 + 6) 2. 8(4n + 9) 3. (x 2)( 6) Identify the terms, like terms, coefficients, and constants in each algebraic expression. 4. v + 7v 2v f + 5 f 6 Terms: Like Terms: Coefficients: Constants: Terms: Like Terms: Coefficients: Constants: Write each expression in simplest form. 6. 4k + 8 9k r + r 8. 4c + 6 2c 10 Add or subtract each linear expression. Show all work!!! Write your answer in simplest form. 9. ( 2x + 5 ) + ( 4x + 12) 10. (7x 4) (x + 6)

24 Test Review Multiple Choice. Show your work next to the problem. Circle your answer! 1. What is the value of 9 2(5 3) + 6? 2. Evaluate 5a + 7b if a = 4 and b = 3? A. 1 C. 20 A. 41 C. 1 B. 11 D. 13 B. 1 D Evaluate 9r+6 s if r = 6 and s = 3? 4. What are the next three terms in the sequence 1.0, 2.5, 4.0, 5.5,..? A. 34 C. 20 A. 7.5, 8.0, 10.5 C. 7.0, 8.5, 10.0 B. 34 D. 20 B. 7.0, 9.5, 11.0 D. 6.0, 7.5, Simplify 4x x Add ( x + 4) + (3x + 2). A. 6x + 15 C. 6x + 1 A. 2x + 6 C. 8x + 6 B. 6x 1 D. 2x + 1 B. 2x + 2 D. 4x + 6

25 7. Subtract ( x 2) (4x + 3). 8. Which expression could be used to find the cost of buying b books and m magazines? School Book Fair Prices Item Cost Magazines $4.95 A. 3x 1 C. 5x 2 A. 7.95b m B. 5x 5 D. 3x 2 B. 7.95m b C. 12.9(b + m) D. 7.95b m Paperback books $ Which expression can be used to find the nth term in this sequence? Figure 1 Figure 2 Figure 3 Figure Number n Number of ? Toothpicks A. n + 5 C. 5n + 1 B. 5n D. 5n Which of the following expressions cannot be factored? A x B. 7x + 3 C. 15x + 10 D. 30x + 40

26 Matching. Write the letter on the line of the matching property illustrated by each expression. You may use a property more than once ( 5 + 3) = 2( 5) + 2(3) A. Commutative Property 12. a + (b + 7) = a + (7 + b) B. Associative Property z = z C. Additive Identity Property 14. 3(x + 4) = 3x + 12 D. Multiplicative Identity Property x + 0 = 10x E. Multiplication Property of Zero 16. 7ab = 7ba F. Distributive Property (0) = (10 + x) + 2x = 10 + (x + 2x) Rewrite each expression in simplest form. Show your work. Circle your answer. 19. (x + 2) + (3x + 1) 20. (5x 7) (3x 4) 21. 8(2x + 5) 22. 2x x x (9x 4) + 3(3x 2)