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By M. Dugopolski

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So the LCD is 2 и 2 и 3 or 12. Now build up each denominator to 12: 3 1 3и3 1и2 ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ ϩ ᎏᎏ Build up each denominator to 12. 4 6 4и3 6и2 9 2 ϭ ᎏᎏ ϩ ᎏᎏ 12 12 11 ϭ ᎏᎏ 12 Simplify. Add. b) The denominators are 12 and 3. Factor 12 as 12 ϭ 2 и 6 ϭ 2 и 2 и 3. Since 3 is a prime number, we do not factor it. Since 2 occurs twice in 12 and not at all in 3, it appears twice in the LCD. Since 3 occurs once in 3 and once in 12, 3 appears once in the LCD. The LCD is 2 и 2 и 3 or 12. So we must build up 1ᎏᎏ to have a 3 denominator of 12: 1 1 1 1и4 ᎏᎏ Ϫ ᎏᎏ ϭ ᎏᎏ Ϫ ᎏᎏ 3 12 3 и 4 12 4 1 ϭ ᎏᎏ Ϫ ᎏᎏ 12 12 3 ϭ ᎏᎏ 12 Build up the first fraction to the LCD.

61. Ϫ16, 9 5 9 63. Ϫᎏᎏ, Ϫᎏᎏ 2 4 62. Ϫ12, Ϫ7 5 6 64. ᎏᎏ, ᎏᎏ 8 7 Miscellaneous Write the interval notation for the interval of real numbers shown in each graph. 91. 2 3 4 5 6 7 8 9 92. Ϫ6 Ϫ4 Ϫ2 0 2 4 6 93. Ϫ40 Ϫ30 Ϫ20 Ϫ10 0 10 Ϫ50 Ϫ40 Ϫ30 Ϫ20 Ϫ10 0 94. 2 d) Explain how to find a rational number between any two given rational numbers. 95. 0 10 20 30 40 50 20 30 13 Fractions 96. Ϫ10 0 10 104. Discussion Suppose that a is a negative real number. Determine whether each of the following is positive or negative, and explain your answer.

12 6 1 2 1 7. ᎏᎏ и ᎏᎏ ϭ ᎏᎏ 2 3 3 1 3 3 8. ᎏᎏ и ᎏᎏ ϭ ᎏᎏ 2 5 10 1 1 9. ᎏᎏ Ϭ 3 ϭ ᎏᎏ 2 6 1 10. 5 Ϭ ᎏᎏ ϭ 10 2 1 1 2 11. ᎏᎏ ϩ ᎏᎏ ϭ ᎏᎏ 2 4 6 1 3 12. 2 Ϫ ᎏᎏ ϭ ᎏᎏ 2 2 Exercises U Study Tips V • Get to know your fellow students. If you are an online student, ask your instructor how you can communicate with other online students. • Set your goals, make plans, and schedule your time. Before you know it, you will have the discipline that is necessary for success. U1V Equivalent Fractions U2V Multiplying Fractions Build up each fraction or whole number so that it is equivalent to the fraction with the indicated denominator.

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