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Chapter 2 Lesson 1 Using Integers and Rational Numbers Objective: You will graph and compare positive and negative numbers. EXAMPLE 1 Graph and compare integers Graph – 3 and – 4 on a number line. Then tell which number is greater. ANSWER On the number line, – 3 is to the right of – 4. So, –3 > – 4. GUIDED PRACTICE for Example 1 Graph the numbers on a number line. Then tell which number is greater. 1. 4 and 0 0 –6 –5 –4 –3 –2 –1 0 4 1 2 3 4 ANSWER On the number line, 4 is to the right of 0. So, 4 > 0. 5 6 GUIDED PRACTICE 2. for Example 1 2 and –5 –5 –6 –5 2 –4 –3 –2 –1 0 1 2 3 4 5 ANSWER On the number line, 2 is to the right of –4. So, 2 > –5. 6 GUIDED PRACTICE 3. for Example 1 –6 and –1 –1 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 ANSWER On the number line, –1 is to the right of –6. So, –1 > –6. EXAMPLE 2 Classify numbers Tell whether each of the following numbers is a whole number, an integer, or a rational number: 5, 0.6, –2 2 and – 24. 3 Integer? Rational Number Whole number? number? 5 Yes Yes Yes 0.6 2 –2 3 –24 No No Yes No No Yes No Yes Yes EXAMPLE 3 Order rational numbers ASTRONOMY A star’s color index is a measure of the temperature of the star. The greater the color index, the cooler the star. Order the stars in the table from hottest to coolest. Star Color index Rigel –0.03 Arneb 0.21 Denebola 0.09 Shaula – 0.22 SOLUTION Begin by graphing the numbers on a number line. EXAMPLE 3 Order rational numbers Read the numbers from left to right: – 0.22, – 0.03, 0.09, 0.21. ANSWER From hottest to coolest, the stars are Shaula, Rigel, Denebola, and Arneb. GUIDED PRACTICE for Examples 2 and 3 Tell whether each numbers in the list is a whole number, an integer, or a rational number.Then order the numbers from least list to greatest. 4. 3, –1.2, –2,0 Number Whole number? Integer? Rational number? 3 Yes Yes Yes –1.2 No No Yes –2 No Yes Yes 0 Yes Yes Yes GUIDED PRACTICE for Examples 2 and 3 ANSWER –2, –1.2, 0, 3. (Ordered the numbers from least to greatest). for Examples 2 and 3 GUIDED PRACTICE 5. 4.5, – 3 , – 2.1, 0.5 4 Number Whole number? Integer? Rational number? 4.5 No No Yes – 3 No No Yes –2 .1 No No Yes 0.5 No No Yes 4 ANSWER – 2.1, – 3 ,0.5 ,– 2.1.(Order the numbers from least to 4 greatest). for Examples 2 and 3 GUIDED PRACTICE 6. 3.6, –1.5,–0.31, – 2.8 Number Whole number? Integer? Rational number? 3.6 No No Yes –1.5 No No Yes –0.31 No No Yes –2.8 No No Yes ANSWER –2.8, –1.5, – 0.31, 3.6 (Ordered the numbers from least to greatest). for Examples 2 and 3 GUIDED PRACTICE 7. 1 ,1.75, – 2 ,0, 3 6 Number Whole number? Integer? Rational number? 1 6 No No Yes No No Yes No No Yes Yes Yes Yes 1.75 – 2 3 0 ANSWER – 2, 0 , 1 , 1.75. (Order the numbers from least to greatest). 3 6 EXAMPLE 4 a. Find opposites of numbers If a = – 2.5, then – a = –(– 2.5) = 2.5. b. If a = 3 , then – a = – 3 . 4 4 EXAMPLE 5 a. Find absolute values of numbers 2 2 2 2 If a = – , then |a | = | | = – ( ) = 3 3 3 3 b. If a = 3.2, then |a| = |3.2| = 3.2. EXAMPLE 6 Analyze a conditional statement Identify the hypothesis and the conclusion of the statement “If a number is a rational number, then the number is an integer.” Tell whether the statement is true or false. If it is false, give a counterexample. SOLUTION Hypothesis: a number is a rational number Conclusion: the number is an integer The statement is false. The number 0.5 is a counterexample, because 0.5 is a 0 rational number but not an integer. GUIDED PRACTICE for Example 4, 5 and 6 For the given value of a, find –a and |a|. 8. a = 5.3 SOLUTION If a = 5.3, then –a = – (5.3) = – 5.3 |a| = |5.3| = 5.3 GUIDED PRACTICE 9. for Example 4, 5 and 6 a=–7 SOLUTION If a = – 7, then –a = – (– 7) = 7 |a| = | – 7| = 7 a= – 4 9 SOLUTION 4 If a = – 4 , then –a = – ( – 4 ) = 9 9 9 |a| = | – 4 | = – ( – 4 ) = 4 9 9 9 10. GUIDED PRACTICE for Example 4, 5 and 6 Identify the hypothesis and the conclusion of the statement. Tell whether the statement is true or false. If it is the false, give a counterexample. 11. If a number is a rational number, then the number is positive SOLUTION Hypothesis: a number is a rational number Conclusion: the number is positive which is false Counterexample: The number –1 is rational, but not positive. GUIDED PRACTICE for Example 4, 5 and 6 12. If a absolute value of a number is a positive, then the number is positive SOLUTION Hypothesis: the absolute value of a number is positive Conclusion: the number is positive which is false false Counter example: the absolute value of –2 is 2 but –2 negative..