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Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc. 1.2 Algebraic Expressions and Sets of Numbers Algebraic Expressions 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. 3x 5 5a 6a 40 2 y y 15 4y 4 2 Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result. Example Evaluate each expression for the given value. a) 5x – 2 for x = 8 = 5(8) – 2 = 40 – 2 = 38 b) 3a2 + 2a + 4 for a = – 4 = 3(– 4)2 + 2(– 4) + 4 = 3(16) + (– 8) + 4 = 44 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 The Number Line A number line is a line on which each point is associated with a number. –5 –4 –3 –2 –1 – 4.8 Negative numbers 0 1 2 3 4 1.5 Positive numbers 5 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. Set Builder Notation A set can also be written in set builder notation. This notation describes the members of a set, but does not list them. Example: { x | x is an even natural number less than 10} The set of such all x that x is an even natural number less than 10 Example Given the set of numbers 1 3 1 , 2.5, , 3.75, 15, 4 2 2, 0, 5 list the numbers in this set that belong to the set of: a. b. c. d. e. f. Natural numbers: 15 Whole numbers: 0, 15 Integers: 0, 15, 5 1 3 Rational numbers: 0, 15, 5, 1 , 3.75, 2.5, 4 2 Irrational numbers: 2 Real numbers: all the numbers in the set 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 |5| = 5 Symbol for absolute value Distance of 4 Distance of 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 Example Find each absolute value. a. 9 9 b. 6 6 c. 4 4 5 5 d. 0 0 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. Example Write the opposite of each number. a. 12 The opposite of 12 is 12. 2 b. 3 The opposite of 2/3 is 2/3. c. 10.4 The opposite of 10.4 is 10.4. Translating Phrases Addition (+) Subtraction (–) sum plus added to more than increased by total difference minus subtract less than decreased by less Multiplicatio n (·) product times multiply multiplied by of double/triple Division () quotient divide shared equally among divided by divided into 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 x 12 x or 12 Example 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