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Section 2.1 Real #s and Absolute Value Algebra I Pass Skills 1.2a Obj: Compare real #s Obj: simplify expressions involving opposites and absolute value. Vocab absolute value the distance of a # from 0.The symbol |x| means the absolute value of x. Integers …-3,-2,-1,0,1,2,3,…. irrational numbers neither repeating or terminating decimal natural numbers 1,2,3,4,…. negative numbers < 0 Opposites opposite sides of a # line. positive numbers >0 rational numbers a/b, fractions real numbers rational & irrational #’s Whole 0,1,2,3,… Compare values of real numbers. Insert an ordering symbol to make each statement true. a. 5 –12 b. 9 a. 5 is to the right of –12 on a number line, so 5 > –12. b. 9 is to the left of 17 on a number line, so 9 < 17. 17 Find the absolute value of a number. To find the absolute value of –11, find the distance from –11 to 0 on a number line. |–11| = 11 Section 2.2 Adding Real #s Pass Skills1 Obj: add #s with like signs. Obj: add #s with unlike signs. Vocab Additive Inverse Property for every real # a there is exactly on real # -a additive inverses any # plus its opposite = 0 Identity Property for Addition any # + 0 = that number Rules for Adding Two Signed Numbers A. Like signs: Find the sum of the absolute values. Use the sign common to both numbers. B. Unlike signs: Find the difference of the absolute values. Use the sign of the number with the greater absolute value. Identity Property for Addition For all real numbers a, a + 0 = a and 0 + a = a. Additive Inverse Property a + (–a) = 0 and –a + a = 0 Add two or more real numbers. Example 1: –15 + (–37) –15+–37= 52 –15 + (–37) = –52 Algebra I Example 2: 16 + (–29) –29–16= 13 –29>16 16 + (–29) = –13 Section 2.3 Subtracting Real #s Algebra I Pass Skills 1 Obj: define subtraction in terms of addition. Obj: subtract Is with llike and unlike signs. Vocab Closure property a set of #s is said to be closed, or to have closure, under a given operation if the result of the operation on any 2 #s in the set is also in the set. Subtraction For all real #s a and b, a-b=a+(-b) Subtract real numbers. To subtract a number, add its opposite. Example 1: –51 – 17 = –51 +(–17) = –68 Example 2: –22 – (–3) = –22 + 3 = –19 Section 2.4 Multiplying and Dividing Real #s Algebra I Pass Skills 1 Obj: multiply and divide positive and negative #s. Obj: define the properties of zero. Vocab Identity property for multiplication for all real #s a, a x 1 = a. Multiplicative inverse reciprocal Multiplicative inverse property for every nonzero real # a, there is exactly one # 1/a such that a x 1/a =1 and 1/a x a =1 Properties of zero let a represent any #. 1. The product of any # and zero is zero 2. Zero divided by any nonzero # is zero 3. Division by zero is undefined. (not possible) Rules and Properties Multiplying Two Numbers With Like Signs (+) (+) = (+) (–) (–) = (+) Multiplying Two Numbers With Unlike Signs (+) (–) = (–) (–) (+) = (–) Dividing Two Numbers With Like Signs (+) ÷ (+) = (+) (–) ÷ (–) = (+) Dividing Two Numbers With Unlike Signs (+) ÷ (–) = (–) (–) ÷ (+) = (–) Multiply and divide real numbers. Find each product or quotient. a. (–25)(–3) (–) (–) = (+) |–25| |–3| (–25)(–3) = 75 b. 128 ÷ (–8) (+) ÷ (–) = (–) |128| ÷ |–8| 128 ÷ (–8) = –16 Section 2.5 Properties and Mental Computation Algebra I Pass Skills 1 Obj: State and apply the commutative, associative, and distributive properties. Obj: use the commutative, associative, and distributive properties to perform mental computations. Vocab Commutative Property of Addition a+b=b+a Commutative Property of Multiplication ab=ba Associative Property of Addition (a + b) + c = a + (b + c) Associative Property of Multiplication (a b) c = a (b c) Distributive Property of Multiplication over Addition and Subtraction: For all numbers a, b, and c. a(b + c) = ab + ac (b + c)a = ba + ca a(b – c) = ab – ac (b – c)a = ba – ca Reflexive Property of Equality: a = a (A number is equal to itself.) Symmetric Property of Equality: If a = b, then b = a. Transitive Property of Equality: If a = b and b = c, then a = c. Substitution Property of Equality: If a = b, then a and b are interchangeable. Simplify: a. (37 + 89) + 11 Use the Associative Property. 37 + (89 + 11) 37 + 100 137 b. (25 13) 4 Use the Commutative Property. (13 25) 4 Use the Associative Property. 13 (25 4) 13 100 1300 To find the opposite of an expression multiply everything by a -1. Section 2.6 Adding and Subtracting Expressions Algebra I Pass Skills 1.2b Obj: use the distributive property to combine like terms. Obj: simplify expressions with several variables. Vocab Coefficient a # that is multiplied by a variable. Like terms terms that contain the same form of a variable. Simplified an algebraic expression is said to be simplified when all of the like terms have been combined and all parentheses have been removed. Term each # in a # sequence; a #, a variable, or a product or quotient of #s and variables that is added or subtracted in an algebraic expression. Add and subtract expressions, and combine like terms. Add (5a + 7b) + (7a +12b). 5a + 7b + 7a + 12b (5a + 7a) + (7b + 12b) 12a + 19b Add and subtract expressions, and combine like terms. Subtract (15x + 8) – (9x + 4). 15x + 8 + (–9x – 4) [15x + (–9x)] + [8 + (–4)] 6x + 4 Section 2.7 Multiplying and Dividing Expressions Algebra I Pass Skills 1.2b Obj: multiply expressions containing variables. Obj: divide expressions containing variables. Vocab Base (of a power) the # that is raised to an exponent. In an expression of the form x2, x is the base. Exponent in a power, the # that tells how many times the base is used as a factor. In an expression of the form xa, a is the exponent. Power an expression of the form xa, where x is the base and a is the exponent. Dividing an Expression Multiply and divide expressions. a. –3(2z + 7) –3(2z) + (–3)(7) –6z – 21 b. (5x + 9) – 2(x + 3) (5x + 9) –2(x) + (–2)(3) 5x + 9 – 2x – 6 3x + 3 4k2 + k + 9