By Mark Dugopolski

ISBN-10: 0073533521

ISBN-13: 9780073533520

Algebra for students, 4th variation, is designed to supply scholars with the algebra historical past wanted for additional college-level arithmetic classes. The unifying subject matter of this article is the advance of the talents useful for fixing equations and inequalities, by way of the appliance of these talents to fixing utilized difficulties. this article comprises 2 chapters, Polynomial & Rational features, and Counting & likelihood, as well as these present in Dugopolski's Intermediate Algebra.

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**Sample text**

D) A ʝ B ͕2͖ because A ʝ B ϭ ͕2, 3͖. 1 Sets 5 U4V Subsets B If every member of set A is also a member of set B, then we write A ʕ B and say that A is a subset of B. See Fig. 4. For example, A ͕2, 3͖ ʕ ͕2, 3, 4͖ because 2 ʦ ͕2, 3, 4͖ and 3 ʦ ͕2, 3, 4͖. Note that the symbol for membership (ʦ) is used between a single element and a set, whereas the symbol for subset (ʕ) is used between two sets. If A is not a subset of B, we write A B. 4 CAUTION To claim that A B, there must be an element of A that does not belong to B.

C) ͕1, 2, 3͖ ͕2, 4, 6, 8͖ d) ͕2, 6͖ ʕ ͕1, 2, 3, 4, 5͖ e) л ʕ ͕2, 4, 6͖ Solution U Helpful Hint V The symbols ʕ and ʚ are often used interchangeably. The symbol ʕ combines the subset symbol ʚ and the equal symbol ϭ. We use it when sets are equal, {1, 2} ʕ {1, 2}, and when they are not, {1} ʕ {1, 2}. When sets are not equal, we could simply use ʚ, as in {1} ʚ {1, 2}. a) True, because 1, 2, and 3 are natural numbers. b) True, because 5, for example, is a natural number and 5 ͕1, 2, 3͖. c) True, because 1 is in the first set but not in the second.

Exploration 99. (5, 7) ʕ R Find the decimal representations of 100. 3 ʦ (6, 7) 2 ᎏᎏ, 9 101. 0 ʦ (0, 1) 102. (0, 2) ʕ W 23 ᎏᎏ, 999 23 ᎏᎏ, 99 234 ᎏᎏ, 999 23 ᎏᎏ, and 9999 1234 ᎏᎏ. 9999 a) What do these decimals have in common? b) What is the relationship between each fraction and its decimal representation? 103. (0, 1) ʝ [1, 2] ϭ л 104. 3 In This Section 2 ᎏᎏ, 99 Operations on the Set of Real Numbers Computations in algebra are performed with positive and negative numbers. In this section, we will extend the basic operations of arithmetic to the negative numbers.

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