By Gillian M. Simpson, Gary C. W. England, Mike J. Harvey

Issues of breeding of canine and cats and the problems of whelping and kittening are mostly handled by means of veterinary surgeons in perform. good fortune is determined by an entire appreciation of the conventional occasions interested in copy, and the issues which can come up. The overseas individuals of this publication describe the traditional physiological foundation of fertility and replica, and the method of difficulties noticeable in perform.

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The easiest way to solve this is to put u = 2x and obtain a quadratic equation in u. For if u = 2x, and So the equation becomes 32u + 256 25 u + 28 4 (u- 2 )(u- 24 ) u2 - u2 - Therefore u = 2x = 24 , and so x = 4. 2 0 0 0 • INEQUALITIES In Chapter 1 we considered identities and equations and used the algebraic properties of the real numbers to solve some of them. When we come to examine inequalities we need some further algebraic rules. First, though, we describe the notation. We write a > b if and only if the number a is greater than the number b.

5 X 4 X 3 X 2 X 1 = 120 One way to think of n! is that it is the nurober of ways in which we can arrange n books on a shelf. , which we define tobe 1, for if there are no books, there is only one way to arrange thero and that is to leave the shelf eropty! Using the factorial syrobol we can now define another syrobol which is known as the syrobol for the binoroial coefficients. Suppose that n and r are both positive integers and that r is less than or equal to n. We define ( ; ) - (n _n;)! r! Using this notation we see that 3!

So the + x) 12 is ( 12 ) = 12! = 12 X 11 X 10 X 9 X 8 = 792 7 5! 7! 5X4X3X2X1 • Ifwe glance back at Pascal's triangle we observe that each row appears to be symmetrical about its midpoint. In other words reading from left to right or reading from right to left seems to give an identical sequence of numbers. We have previously observed that each entry in the table appears to be the sum of two entries in the row above. These are two significant features of the binomial coefficients which we can now prove.