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Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
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1.2 Algebraic Expressions and Sets of Numbers
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A variable is a letter used to represent any number. A constant is either a fixed number or a letter that represents a fixed number. An algebraic expression is formed by numbers and variables connected by the operation of addition, subtraction, multiplication, division, raising to powers, and/or taking roots. Algebraic Expressions
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Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result.
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Example Evaluate each expression for the given value. a) 5x – 2 for x = 8 = 5(8) – 2 = 40 – 2 = 38 b) 3a 2 + 2a + 4 for a = – 4 = 3(– 4) 2 + 2(– 4) + 4 = 3(16) + (– 8) + 4 = 44
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Set of Numbers Natural numbers – {1, 2, 3, 4, 5, 6...} Whole numbers – {0, 1, 2, 3, 4...} Integers – {... –3, -2, -1, 0, 1, 2, 3...} Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator 0 Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers Real numbers – the set of all rational and irrational numbers combined
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The Number Line A number line is a line on which each point is associated with a number. 2– 201345– 1– 3– 4– 5 Negative numbers Positive numbers – 4.81.5
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Vocabulary Variable is a symbol used to represent a number. Algebraic expressions are a collection of numbers, variables, operations, grouping symbols, but NO = or inequalities. In describing some of the previous sets, we used a... symbol, called an ellipsis. It means to continue in the same pattern. The members of a set are called its elements. When we list (or attempt to list with an ellipsis) the elements of a set, the set is written in roster form.
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A set can also be written in set builder notation. This notation describes the members of a set, but does not list them. { x | x is an even natural number less than 10} The set of all x such that x is an even natural number less than 10 Set Builder Notation Example:
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Example Given the set of numbers list the numbers in this set that belong to the set of: a. Natural numbers: b. Whole numbers: c. Integers: d. Rational numbers: e. Irrational numbers: f. Real numbers:
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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. 2– 201345– 1– 3– 4– 5 | – 4| = 4 Distance of 4 Symbol for absolute value |5| = 5 Distance of 5
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Example Find each absolute value. a. b. c. d.
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Finding the Opposite of a Number Opposite The opposite of a number a is the number a. Double Negative Property For every real number a, ( a) = a.
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Example Write the opposite of each number. a. 12 The opposite of 12 is 12. b. The opposite of 2/3 is 2/3. c. 10.4 The opposite of 10.4 is 10.4.
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Translating Phrases Addition (+) Subtraction (–) Multiplicatio n (·) Division ( ) sum plus added to more than increased by total difference minus subtract less than decreased by less product times multiply multiplied by of double/triple quotient divide shared equally among divided by divided into
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Example Write as an algebraic expression. Let x represent the unknown number. a. 5 decreased by a number 5 – x b. The quotient of a number and 12
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Translate the following phrases into algebraic expressions. a. The sum of 4 and twice y 4 + 2y b. x less than the product of y and z yz x Example
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