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By Sheldon Axler

University Algebra and Trigonometry will entice those that are looking to supply vital subject matters extra in-depth, higher-level insurance. this article deals streamlined strategy followed with available definitions throughout all chapters to permit for an easy-to-understand learn. collage Algebra comprises prose that's special, actual, and simple to learn, with basic definitions of even the themes which are regularly such a lot tricky for readers.

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2 Algebra of the Real Numbers 23 solution solution y −4 2 5 y −4 x+3 2 + = · + · x+3 5 x+3 5 5 x+3 39. y 1 x − x−y y x 2 · 5 + (y − 4)(x + 3) 5(x + 3) = x2 − y 2 1 x−y xy = 10 + yx + 3y − 4x − 12 5(x + 3) = 1 (x + y)(x − y) x−y xy = xy − 4x + 3y − 2 5(x + 3) = x+y xy 4t + 1 3 + t2 t 45. (x + a)2 − x 2 a solution 4t + 1 3 4t + 1 3 t + = + · t2 t t2 t t = 4t + 1 3t + 2 t2 t = 7t + 1 t2 3 v+1 + v(v − 2) v3 (x + a)2 − x 2 x 2 + 2xa + a2 − x 2 = a a = 47. 1 x y − x−y y x 3v2 v2 − v − 2 + 3 v3 (v − 2) v (v − 2) = 4v2 − v − 2 v3 (v − 2) x−2 y z x+2 solution 3 v+1 v2 3 v+1 v−2 + = 2 · + · 3 v(v − 2) v v v(v − 2) v3 v−2 = 2xa + a2 a = 2x + a solution 43.

You should begin to get comfortable working in such situations. To get a good understanding of an abstract piece of mathematics, start by looking at an example using concrete numbers, as in Example 5, before going on to a more abstract setting, as in Example 6. Suppose b is a real number and h > 0. (a) Write the inequality |x − b| < h without using an absolute value. (b) Write the set {x : |x − b| < h} as an interval. solution (a) The absolute value of a number is less than h precisely when the number is between −h and h.

To see why transitivity holds, suppose a < b and b < c. Then a is left of b on the real line and b is left of c. This implies that a is left of c, which means that a < c; see the figure below. a b For example, from the inequalities √ 15 < 4 and 4 < 21 5 we can conclude that √ 15 < 21 . 5 c Transitivity: a < b and b < c implies that a < c. Often multiple inequalities are written together as a single string of inequalities. Thus a < b < c means the same thing as a < b and b < c. Our next result shows that we can add inequalities.

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