Modern Engineering Mathematics (5th Edition) by Glyn James

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2 N UMBE R AN D ARI TH ME TIC 11 where the comparator у indicates ‘greater than or equal to’. ) Geometrically | x | is the distance of the point representing x on the number line from the point representing zero. Similarly | x − a | is the distance of the point representing x on the number line from that representing a. The set of numbers between two numbers, a and b say, defines an open interval on the real line. This is the set {x:a Ͻ x Ͻ b, x in ‫ }ޒ‬and is usually denoted by (a, b). ) Here the double-sided inequality means that x is greater than a and less than b; that is, the inequalities a Ͻ x and x Ͻ b apply simultaneously.

Find the values of x such that x 2 + 2x + 2 Ͼ 50 Solution Completing the square on the left-hand side of the inequality we obtain (x + 1)2 + 1 Ͼ 50 which gives (x + 1)2 Ͼ 49 Taking the square root of both sides of this inequality we deduce that either (x + 1) Ͻ −7 or (x + 1) Ͼ 7 Note particularly the first of these inequalities. From these we deduce that x 2 + 2x + 2 Ͼ 50 for x Ͻ −8 or x Ͼ 6 The reader should check these results using spot values of x, say x = −10 and x = 10. 26 Solution .. A food manufacturer found that the sales figure for a certain item depended on its selling price.

5⎞ ⎛ 5⎞ ⎛ 5⎞ = 1, ⎜ ⎟ = = 5, ⎜ ⎟ = = 10 and so on. 5! 4! 3! 7 33 Exercises Given a 0 = 2, a1 = −1, a 2 = −4, a3 = 5, a4 = 3 and b0 = 1, b1 = 1, b2 = 2, b3 = −1, b4 = 2, calculate 4 (a) ∑ ak 3 (b) k=0 ∑ ak bk k =1 .. ∑ ai i =1 Evaluate (a) 5! /4! /(3! 14. Conventionally the point P is represented by an ordered pair of numbers contained in brackets thus: (x, y). This innovation was largely due to Descartes and consequently we often refer to (x, y) as the cartesian coordinates of P. 6, but has an entirely separate meaning and the two should not be confused.