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57328-0136147054_App - Appendix A A.1 |3 17| = | 14| =(14 =...

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Appendix A A.1: | 3 17 | = | − 14 | = ( 14) = 14. A.2: |− 3 | + | 17 | = ( 3) + 17 = 3 + 17 = 20. A.3: 0 . 25 1 4 = 1 2 = 1 2 = 1 2 . A.4: | 5 | − |− 7 | = 5 7 = 2. A.5: | ( 5)(4 9) | = | ( 5)( 5) | = | 25 | = 25. A.6: |− 6 | | 4 | + |− 2 | = 6 4 + 2 = 1. A.7: | ( 3) 3 | = |− 27 | = 27. A.8: 3 3 = 3 3 because 3 > 3 . A.9: According to Mathematica , π 3 . 14159265358979323846264338327950288419716939937510582097494 , whereas 22 7 3 . 14285714285714285714285714285714285714285714285714285714286 . Therefore π 22 7 < 0 , and thus π 22 7 = 22 7 π. For a more elegant proof of the inequality, see Miscellaneous Problems 110 and 111 of Chapter 7. A.10: −| 7 4 | = (7 4) = 3. A.11: If x < 3 then x 3 < 0, so in this case | x 3 | = ( x 3) = 3 x . A.12: Given | x 7 | < 1, it follows from the fourth property of absolute value in (3) that 1 < x 7 < 1, and thus 6 < x < 8. Therefore | x 5 | + | x 10 | = ( x 5) + (10 x ) = 5 . A.13: 2 x 7 < 3: 2 x < 4; x < 2. Solution set: ( −∞ , 2). A.14: 1 4 x > 2: 4 x < 1 2; x < 1 4 . Solution set: ( −∞ , 1 4 ) . A.15: 3 x 4 17: 3 x 21; x 7. Solution set: [7 , + ). 1920
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A.16: 2 x + 5 9: 2 x 4; x 2. Solution set: ( −∞ , 2 ]. A.17: 2 3 x < 7: 3 x < 5; 3 x > 5; x > 5 3 . Solution set: ( 5 3 , + ) . A.18: 6 5 x > 9: 5 x > 15; 5 x < 15; x < 3. Solution set: ( −∞ , 3). A.19: 3 < 2 x + 5 < 7: 8 < 2 x < 2; 4 < x < 1. Solution set: ( 4 , 1). A.20: 4 3 x 5 10: 9 3 x 15; 3 x 5. Solution set: [ 3 , 5 ]. A.21: 6 5 2 x < 2: 11 2 x < 3; 3 < 2 x 11; 3 2 < x 11 2 . Solution set: ( 3 2 , 11 2 . A.22: 3 < 1 5 x < 7: 2 < 5 x < 6; 6 < 5 x < 2; 6 5 < x < 2 5 . Solution set: ( 6 5 , 2 5 ) . A.23: If | 3 2 x | < 5, then by the fourth property of absolute value in (3) we have 5 < 3 2 x < 5. Thus 8 < 2 x < 2, and hence 2 < 2 x < 8. Therefore 1 < x < 4. So the solution set of the original inequality is ( 1 , 4). A.24: If | 5 x + 3 | 4, then by an extension of the fourth property of absolute value in (3) we have 4 5 x + 3 4. Thus 7 5 x 1, and so 7 5 x 1 5 . Thus the solution set of the original inequality is the interval 7 5 , 1 5 . A.25: We will find the complement of the solution set—those real numbers not in the solution set—by solving instead the inequality | 1 3 x | 2. By an extension of the fourth property of absolute value in (3), we then have 2 1 3 x 2. It follows that 3 3 x 1, and hence 1 3 x 3. So the complement of the solution set of the original equality is the closed interval 1 3 , 1 . Therefore the solution set of the inequality | 1 3 x | > 2 is ( −∞ , 1 3 ) (1 , + ). A.26: Given: 1 < | 7 x 1 | < 3. Case 1: 7 x 1 0. This is equivalent to the assertion that x 1 7 ; moreover, the given inequality takes the form 1 < 7 x 1 < 3; that is, 2 < 7 x < 4, so that 2 7 < x < 4 7 . Hence the solutions we obtain in Case 1 are the real numbers in the interval ( 2 7 , 4 7 ) . Case 2: 7 x 1 < 0. This is equivalent to the assertion that x < 1 7 ; moreover, the given inequality takes the form 1 < 1 7 x < 3. Hence 0 < 7 x < 2, so that 2 < 7 x < 0, and therefore 2 7 < x < 0. So the solutions we obtain in Case 2 are the real numbers in the interval ( 2 7 , 0 ) . These two cases are exclusive and exhaustive, and so the solution set of the original inequality is ( 2 7 , 0 ) ( 2 7 , 4 7 ) .
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