46 Chapter 1 The Set of Real Numbers Section 1.1 Introduction to Algebra and the Set Concepts 1. Variables and Expressions 2. The Set of Real Numbers 3. Inequalities 4. Opposite of a Real Number 5. Absolute Value of a Real Number Skill Practice 2. After dining out at a restaurant, the minimum recommended tip for the server is 15% of the cost of the meal. This can be represented by 0.15c, where c is the cost of the meal. Compute the tip for a meal that costs a. $18 b. $46 1. Variables and Expressions Doctors promote daily exercise as part of a healthy lifestyle. Aerobic exercise is exercise for the heart. During aerobic exercise the goal is to maintain a heart rate level between 65% and 85% of an individual’s maximum recommended heart rate. The maximum recommended heart rate, in beats per minute, for an adult of age a is given by Maximum recommended heart rate 220 a In this example, the value a is called a variable. This is a symbol or letter, such as x, y, z, a, and the like, that is used to represent an unknown number that is subject to change. The number 220 is called a constant, because it does not vary. The quantity 220 a is called an algebraic expression. An algebraic expression is a collection of variables and constants under algebraic operations. For example , y z, and t 1.4 are algebraic expressions. 3x The symbols used in algebraic expressions to show the four basic operations are shown here: Addition a b Subtraction a b Multiplication a b, a · b, (a)b, a(b), (a)(b), ab (Note:We rarely use the notation a b because the symbol, , may be confused with the variable x.) Division a b, a b , a/b, ba The value of an algebraic expression depends on the values of the variables within the expression. Evaluating an Algebraic Expression Example 1 The expression 220 a represents the maximum recommended heart rate for an adult of age a. Determine the maximum heart rate for: a. A 20-year-old b. A 45-year-old Solution: a. In the expression 220 a, the variable, a, represents the age of the individual.To calculate the maximum recommended heart rate for a 20-year-old, substitute 20 for a in the expression. 220 a 220 ( ) When substituting a number for a variable, use parentheses. 220 (20) Substitute a 20. 200 Subtract. The maximum recommended heart rate for a 20-year-old is 200 beats per minute. Concept Connections 1. Why is not a variable? Classroom Example: p. 55, Exercise 4 Answers 1. The symbol is not a variable because its value does not vary. It is always equal to the irrational number 3.14159. . . . 2. a. $2.70 b. $6.90 of Real Numbers
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