# The Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,

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1 1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition are more than and increased by. 7 more than a is a 7. b increased by 3 is b 3. p. 54 Subtraction x y means the difference of x and y or x minus y. Some other The difference of x and 3 words indicating subtraction are less than and decreased by. is x 3. 5 less than p is p 5. a decreased by 4 is a 4. p. 54 Multiplication x y (x)(y) These all mean the product of x and y or x times y. xy Division x means x divided by y or the quotient when x is divided by y. y The product of m and n is mn. The product of 2 and the sum of a and b is 2(a b). p. 55 n n divided by 5 is 5. The sum of a and b, divided a b by 3, is. 3 p. 57 The Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers, 1. a b b a a b b a p. 63 The Associative Properties If a, b, and c are any numbers, 1. a (b c) (a b) c 3 (7 12) (3 7) a (b c) (a b) c 2 (5 12) (2 5) 12 p. 63 The Distributive Property If a, b, and c are any numbers, a(b c) a b a c 6 (8 15) p. 65 Adding and Subtracting Signed Numbers Section 1.3 Adding Signed Numbers 1. If two numbers have the same sign, add their absolute values Give the sum the sign of the original numbers. ( 9) ( 7) 16 p If two numbers have different signs, subtract their absolute 15 ( 10) 5 values, the smaller from the larger. Give the sum the sign of ( 12) 9 3 the number with the larger absolute value. p. 73 Continued 131

2 132 CHAPTER 1 THE LANGUAGE OF ALGEBRA DEFINITION/PROCEDURE EXAMPLE REFERENCE Adding and Subtracting Signed Numbers Section 1.3 Subtracting Signed Numbers 1. Rewrite the subtraction problem as an addition problem by ( 8) a. Changing the subtraction symbol to an addition symbol 8 b. Replacing the number being subtracted with its opposite ( 15) 2. Add the resulting signed numbers as before. 7 9 ( 7) p. 77 Multiplying and Dividing Signed Numbers Section 1.4 Multiplying Signed Numbers Multiply the absolute values of the two numbers. 5( 7) If the numbers have different signs, the product is negative. ( 10)(9) 90 p If the numbers have the same sign, the product is positive ( 9)( 8) 72 p. 90 Dividing Signed Numbers Divide the absolute values of the two numbers. 1. If the numbers have different signs, the quotient is negative. 2. If the numbers have the same sign, the quotient is positive p. 92 Evaluating Algebraic Expressions Section 1.5 Algebraic Expressions An expression that contains numbers and letters (called variables). p. 103 Evaluating Algebraic Expressions Evaluate 2x 3y if x 5 and To evaluate an algebraic expression: y Replace each variable or letter with its number value. 2x 3y 2. Do the necessary arithmetic, following the rules for the 2 5 (3)( 2) order of operations p. 103 Adding and Subtracting Terms Section 1.6 Term A number or the product of a number and one or more variables. p. 115 Combining Like Terms 5x 2x 7x To combine like terms: Add or subtract the coefficients (the numbers multiplying the variables). 8a 5a 3a 2. Attach the common variable. 8 5 p. 117 Multiplying and Dividing Terms Section 1.7 Property 1 of Exponents a m a n a m n p. 123 Property 2 of Exponents a m a n a m n p.125

3 Summary Exercises This exercise set is provided to give you practice with each of the objectives of the chapter. Each exercise is keyed to the appropriate chapter section. The answers are provided in the Instructor s Manual. Your instructor will give you guidelines on how to best use these exercises. [1.1] Write, using symbols more than y 2. c decreased by The product of 8 and a 4. The quotient when y is divided by times the product of m and n 6. The product of a and 5 less than a 7. 3 more than the product of 17 and x 8. The quotient when a plus 2 is divided by a minus 2 Identify which are expressions and which are not. 9. 4(x 3) y (3x 9) [1.2] Identify the property that is illustrated by each of the following statements (7 12) (5 7) (8 3) (5 3) (4 5) Verify that each of the following statements is true by evaluating each side of the equation separately and comparing the results (5 4) (3 7) (7 9) 4 7 (9 4) 20. (2 3) 6 2 (3 6) 21. (8 2) 5 8(2 5) 22. (3 7) 2 3 (7 2) Use the distributive law to remove parentheses (7 4) 24. 4(2 6) 25. 4(w v) 26. 6(x y) 27. 3(5a 2) 28. 2(4x 2 3x) [1.3] Add ( 8) ( 4) ( 6) ( 16) ( 9.7) ( 3)

4 134 CHAPTER 1 THE LANGUAGE OF ALGEBRA Subtract ( 7) ( 1) ( 9) ( 2) ( 8.1) Find the median for each of the following sets , 4, 9, 10, , 3, 2, 4, , 8, 4, 1, , 3, 2, 5, , 4, 1, 8, 6, , 1, 5, 3, 4, 1 Determine the range for each of the following sets , 5, 1, 8, , 5, 6, 4, 2, , 2, 1, 3, , 3, 5, 3, 4 [1.4] Multiply. 55. (10)( 7) 56. ( 8)( 5) 57. ( 3)( 15) 58. (1)( 15) 59. (0)( 8) ( 4) 62. Divide ( 1) Perform the indicated operations ( 10) 5 ( 2) Evaluate each of the following expressions ( 2) (18 3) (5 4) (3 2 4) 78. 5(4 2) (5 4 2) (5 2) (3 5 2) 2

5 SUMMARY EXERCISES 135 [1.5] Evaluate the expressions if x 3, y 6, z 4, and w x w 85. 5y 4z 86. x y 3z 87. 5z x 2 2w x 3 6z 2x 4z 90. 5(x 2 w 2 ) w y z 3x y x(y 2 z 2 ) y(x w) w x (y z)(y z) x 2 2xw w 2 [1.6] List the terms of the expressions a 3 3a x 2 7x 3 Circle like terms m 2, 3m, 4m 2, 5m 3, m ab 2, 3b 2, 5a, ab 2, 7a 2, 3ab 2, 4a 2 b Combine like terms c 7c x 5x a 2a c 3c xy 6xy ab 2 2ab a 3b 12a 2b x 2x 5y 3x x 3 17x 2 2x 3 8x a 3 5a 2 4a 2a 3 3a 2 a 110. Subtract 4a 3 from the sum of 2a 3 and 12a Subtract the sum of 3x 2 and 5x 2 from 15x 2. [1.7] Divide x 10 a x 3 a x 2 x m 2 m 3 m 4 x p 7 24x p 5 8x m 7 n 5 108x 9 y m 2 n 3 9xy p 5 q 3 52a 5 b 3 c p 3 q 13a 4 c 122. (4x 3 )(5x 4 ) 123. (3x) 2 (4xy) 124. (8x 2 y 3 )(3x 3 y 2 ) 125. ( 2x 3 y 3 )( 5xy) 126. (6x 4 )(2x 2 y) m 5

6 136 CHAPTER 1 THE LANGUAGE OF ALGEBRA Write the algebraic expression that answers the question. [ ] 127. Carpentry. If x feet (ft) are cut off the end of a board that is 23 ft long, how much is left? 128. Money. Joan has 25 nickels and dimes in her pocket. If x of these are dimes, how many of the coins are nickels? 129. Age. Sam is 5 years older than Angela. If Angela is x years old now, how old is Sam? 130. Money. Margaret has \$5 more than twice as much money as Gerry. Write an expression for the amount of money that Margaret has Geometry. The length of a rectangle is 4 meters (m) more than the width. Write an expression for the length of the rectangle Number problem. A number is 7 less than 6 times the number n. Write an expression for the number Carpentry. A 25-ft plank is cut into two pieces. Write expressions for the length of each piece Money. Bernie has x dimes and q quarters in his pocket. Write an expression for the amount of money that Bernie has in his pocket.

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### ( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

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### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b, MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a