Chapter 8 - XYZ Custom Plus

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450<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsIn algebra we can use inequality symbols when comparing numbers.NotationIf a and b are any two numbers on the number line, thena , b is read “a is less than b”a . b is read “a is greater than b”Practice ProblemsWrite each statement in words.1. 2 , 82. 5 . 10 (Is this a true statement?)As you can see, the inequality symbols always point to the smaller of the twonumbers being compared. Here are some examples that illustrate how we use theinequality symbols.Example 13 , 5 is read “3 is less than 5.” Note that it would also becorrect to write 5 . 3. Both statements, “3 is less than 5” and “5 is greater than 3,”have the same meaning. The inequality symbols always point to the smaller number.Example 20 . 100 is a false statement, because 0 is less than 100,not greater than 100. To write a true inequality statement using the numbers 0and 100, we would have to write either 0 , 100 or 100 . 0.3. 24 , 4Example 323 , 5 is a true statement, because 23 is to the left of 5on the number line, and, therefore, it must be less than 5. Another statement thatmeans the same thing is 5 . 23.4. 27 , 22Example 425 , 22 is a true statement, because 25 is to the left of22 on the number line, meaning that 25 is less than 22. Both statements25 , 22 and 22 . 25 have the same meaning; they both say that 25 is a smallernumber than 22.BAbsolute ValueIt is sometimes convenient to talk about only the numerical part of a numberand disregard the sign (1 or 2) in front of it. The following definition gives us away of doing this.DefinitionGive the absolute value of each ofthe following.5. u6u6. u25uAnswers1. 2 is less than 8.2. 5 is greater than 10. (No.)3. 24 is less than 4.4. 27 is less than 22.5. 6 6. 5The absolute value of a number is its distance from 0 on the number line.We denote the absolute value of a number with vertical lines. For example,the absolute value of 23 is written u23u.The absolute value of a number is never negative because it is a distance, and adistance is always measured in positive units (unless it happens to be 0).Here are some examples of absolute value problems.Example 5u5u 5 5 The number 5 is 5 units from 0.Example 6u 23u 5 3 The number 23 is 3 units from 0.
8.1 Positive and Negative Numbers451Example 7u 27u 5 7 The number 27 is 7 units from 0.Give the absolute value.7. u28uCOppositesDefinitionTwo numbers that are the same distance from 0 but in opposite directionsfrom 0 are called opposites.* The notation for the opposite of a is 2a.Example 8Give the opposite of each of the following numbers:5, 7, 1, 25, 288. Give the opposite of each of thefollowing numbers: 8, 10, 0, 24.Solution The opposite of 5 is 25.The opposite of 7 is 27.The opposite of 1 is 21.The opposite of 25 is 2(25), or 5.The opposite of 28 is 2(28), or 8.We see from this example that the opposite of every positive number is a negativenumber, and likewise, the opposite of every negative number is a positivenumber. The last two parts of Example 8 illustrate the following property:PropertyIf a represents any positive number, then it is always true that2(2a) 5 aIn other words, this property states that the opposite of a negative number is apositive number.It should be evident now that the symbols 1 and 2 can be used to indicateseveral different ideas in mathematics. In the past we have used them to indicateaddition and subtraction. They can also be used to indicate the direction anumber is from 0 on the number line. For instance, the number 13 (read “positive3”) is the number that is 3 units from zero in the positive direction. On the otherhand, the number 23 (read “negative 3”) is the number that is 3 units from 0 inthe negative direction. The symbol 2 can also be used to indicate the opposite ofa number, as in 2(22) 5 2. The interpretation of the symbols 1 and 2 depends onthe situation in which they are used. For example:3 1 5 The 1 sign indicates addition.7 2 2 The 2 sign indicates subtraction.27 The 2 sign is read “negative” 7.2(25) The first 2 sign is read “the opposite of.” The second 2sign is read “negative” 5.This may seem confusing at first, but as you work through the problems in thischapter you will get used to the different interpretations of the symbols 1 and 2.We should mention here that the set of whole numbers along with their oppositesforms the set of integers. That is:Integers 5 {. . . , 23, 22, 21, 0, 1, 2, 3, . . .}*In some books opposites are called additive inverses.Answers7. 8 8. 28, 210, 0, 4
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Chapter 8 - XYZ Custom Plus

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