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Absolute Value The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. | – 4| = 4 Symbol for absolute value |5| = 5 Distance of 4 –5 –4 –3 –2 –1 Distance of 5 0 1 2 3 4 5 Example Find each absolute value. a. 9 b. 6 c. 4 5 d. 0 Adding Real Numbers To add two real numbers: 1. With the same sign, add their absolute values. Use their common sign as the sign of the answer. 2. With different signs, subtract their absolute values. Give the answer the same sign as the number with the larger absolute value. Example Add. 1. (‒8) + (‒3) 2. (‒7) + 1 3. (‒12.6) + (‒1.7) 9 2 4. 10 10 Subtracting Real Numbers Opposite of a Real number If a is a real number, then –a is its opposite. Subtracting Two Real Numbers If a and b are real numbers, then a – b = a + (– b). Example Subtract. 1. 4 ‒ 7 2. ‒8 ‒ (‒9) 3. (–5) – 6 – (–3) 4. 6.9 ‒ (‒1.8) 3 4 5. 4 5 Multiplying Real Numbers Multiplying Real Numbers 1. The product of two numbers with the same sign is a positive number. 2. The product of two numbers with different signs is a negative number. Examples Multiply. 1. 4(–2) 2. ‒7(‒5) 3. 9(‒6.2) 3 1 4. 4 7 Product Property of 0 a · 0 = 0. Also 0 · a = 0. Example: Multiply. –6 · 0 Example: Multiply. 0 · 125 Quotient of Two Real Numbers • The quotient of two numbers with the same sign is positive. • The quotient of two numbers with different signs is negative. • Division by 0 is undefined. Example Divide. a. 20 4 b. 36 3 c. 56 0.8 Examples a. Find the quotient. b. Find the quotient. 36 12 3 1 2 6 Simplifying Real Numbers If a and b are real numbers, and b 0, a a a b b b Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 · 3 · 3 · 3 • 3 is the base • 4 is the exponent (also called power) Evaluate. a. (–2)4 b. ‒72 The Order of Operations Order of Operations (P)Simplify expressions using the order that follows. If grouping symbols such as parentheses are present, simplify expression within those first, starting with the innermost set. If fraction bars are present, simplify the numerator and denominator separately. (E) Evaluate exponential expressions, roots, or absolute values in order from left to right. (M-D) Multiply or divide in order from left to right. (A-S) Add or subtract in order from left to right. Example Use order of operations to evaluate each expression. a. 7(9) 2(6) d. 6 9 3 3 2 b. 8 3(2) 9 2(3) c. 32 8 5 9 e. (4)2 9 3 2 7 Commutative and Associative Property Commutative property • Addition: a + b = b + a • Multiplication: a · b = b · a Associative property • Addition: (a + b) + c = a + (b + c) • Multiplication: (a · b) · c = a · (b · c) Example Use the commutative or associative property to complete. a. x + 8 = ______ b. 7 · x = ______ c. 3 + (8 + 1) = _________ d. (‒5 ·4) · 2 = _________ e. (xy) ·18 = ___________ Distributive Property For real numbers, a, b, and c. a(b + c) = ab + ac Also, a(b c) = ab ac Example Use the distributive property to remove the parentheses. 7(4 + 2) = 7(4 + 2) = (7)(4) + (7)(2) = 28 + 14 = 42 Example Use the distributive property to write each expression without parentheses. Then simplify the result. a. 3(2x – y) b. -5(‒3 + 9z) c. ‒(5 + x ‒ 2w) Example Write each as an algebraic expression. 1. 2. 3. 4. A vending machine contains x quarters. Write an expression for the value of the quarters. The cost of y tables if each tables costs $230. Two numbers have a sum of 40. If one number is a, represent the other number as an expression in a. Two angles are supplementary if the sum of their measures is 180 degrees. If the measure of one angle is x degrees represent the other angle as an expression in x. Like Terms Terms of an expression are the addends of the expression. Like terms contain the same variables raised to the same powers. Example Simplify each expression. a. 7 x 5 3x 2 b. 3 y 2 y 5 7 y 2 2 c. 7 x 3 5( x 4) 1 1 1 d. (4a 6b) (9a 12b 1) 2 3 4