By Mark Dugopolski

Algebra for students, 4th version, is designed to supply scholars with the algebra historical past wanted for extra college-level arithmetic classes. The unifying subject matter of this article is the advance of the abilities priceless for fixing equations and inequalities, by means of the appliance of these abilities to fixing utilized difficulties. this article includes 2 chapters, Polynomial & Rational capabilities, and Counting & chance, as well as these present in Dugopolski's Intermediate Algebra.

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**Sample text**

0 63. ͕2, 3͖ 65. ͕3, ͙2ෆ ͖ R 62. 0 I R 77. 3 4 5 6 7 8 9 78. 3 4 5 6 7 8 9 Q I Q 64. ͕0, 1͖ Q 76. The set of real numbers greater than 2 and less than or equal to 5. Write the interval notation for the interval of real numbers shown in each graph. Z 60. ͙2 ෆ R 75. The set of real numbers greater than or equal to 1 and less than 3 in each blank so that 45. N ෆ 59. ͙2 72. The set of real numbers between Ϫ1 and 3 66. ͕3, ͙2 ෆ͖ 79. Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 N Q U5V Intervals of Real Numbers Write each interval of real numbers in interval notation and graph it.

25 Now do Exercises 67–76 The intersection of two intervals is the set of real numbers that belong to both intervals. The union of two intervals is the set of real numbers that belong to one, or the other, or both of the intervals. E X A M P L E 6 Combining intervals Write each union or intersection as a single interval. a) (2, 4) ʜ (3, 6) b) (2, 4) പ (3, 6) c) (Ϫ1, 2) ഫ [0, ϱ) d) (Ϫ1, 2) പ [0, ϱ) Solution a) Graph (2, 4) and (3, 6) as in Fig. 26 on the next page. The union of the two intervals consists of the real numbers between 2 and 6, which is written as (2, 6).

The set of real numbers can be visualized as the set of all points on the number line. Two real numbers are equal if they correspond to the same point on the number line. See Fig. 14. 15 illustrates the relationship between the set of real numbers and the various subsets that we have been discussing. 16, 12· are members of each of these sets. a) Real numbers b) Rational numbers c) Integers Solution a) All of the numbers are real numbers. 16, and 12 are rational numbers. c) The only integers in this set are 0 and 12.