27. A person spends one-third of his income, saves one-fourth, and pays away 5 per cent. on the whole as interest at 71 per cent. on debts previously incurred, and then has $110 remaining: what was the amount of his debts? 28. Two vessels contain mixtures of wine and water; in one there is three times as much wine as water, in the other five times as much water as wine. Find how much must be drawn off from each to fill a third vessel which holds seven gallons, in order that its contents may be half wine and half water. 29. There are two mixtures of wine and water, one of which contains twice as much water as wine, and the other three times as much wine as water. How much must there be taken from each to fill a pint cup, in which the water and the wine shall be equally mixed ? 30. Two men set out at the same time to walk, one from A to B, and the other from B to A, a distance of a miles. The former walks at the rate of p miles, and the latter at the rate of q miles an hour: at what distance from A will they meet? 31. A train runs from A to B in 3 hours ; a second train runs from A to C, a point 15 miles beyond B, in 31 hours, travelling at a speed which is less by 1 mile per hour. Find distance from A to B. 32. Coffee is bought at 36 cents and chicory at 9 cents per lb.; in what proportion must they be mixed that 10 per cent. may be gained by selling the mixture at 33 cents per lb. ? 33. A man has one kind of coffee at a cents per pound, and another at b cents per pound. How much of each must he take to form a mixture of a-6 lbs., which he can sell at c cents a pound without loss ? 34. A man spends c half-dollars in buying two kinds of silk at a dimes and b dimes a yard respectively; he could have bought 3 times as much of the first and half as much of the second for the same money. How many yards of each did he buy? 35. A man rides one-third of the distance from A to B at the rate of a miles an hour, and the remainder at the rate of 26 miles an hour. If he had travelled at a uniform rate of 3c miles an hour, he could have ridden from A to B and back again in the same time. Prove 2 1 1 that + b 36. A, B, C are three towns forming a triangle. A man has to walk from one to the next, ride thence to the next, and drive thence to his starting point. He can walk, ride, and drive a mile in a, b, c minutes respectively. If he starts from B he takes a +c-6 hours, if he starts from C he takes b+a-c hours, and if he starts from A he takes c+b- a hours. Find the length of the circuit. a CHAPTER XXV. INEQUALITIES. 192. Any quantity a is said to be greater than another quantity b when a-6 is positive; thus 2 is greater than -3, because 2-(-3), or 5 is positive. Also b is said to be less than a when b-a is negative; thus – 5 is less than – 2, because -5-(-2), or – 3 is negative. In accordance with this definition, zero must be regarded as greater than any negative quantity. 193. The statement in algebraic language that one expression is greater or less than another is called an inequality. 194. The sign of inequality is >, the opening being placed towards the greater quantity. Thus a>b is read “a is greater than b." 195. The first and second members are the expressions on the left and right, respectively, of the sign of inequality. 196. Inequalities subsist in the same sense when corresponding members in each are the greater or the less. Thus the inequalities a>b and 7>5 are said to subsist in the same sense. In the present chapter we shall suppose (unless the contrary is directly stated) that the letters always denote real and positive quantities. 197. If a>b, then it is evident that a+c>b+c; 6 a that is, an inequality will still hold after each side has been increased, diminished, multiplied, or divided by the same positive quantity. 198. If a-c>b, by adding c to each side, a>b+c; which shows that in an inequality any term may be transposed from one side to the other if its sign be changed. 199. If a>b, then evidently b<a; that is, if the sides of an inequality be transposed, the sign of inequality must be reversed. 200. If a>b, then a-b is positive, and b-a is negative; that is, –a-(-6) is negative, and therefore -ax-b; hence, if the signs of all the terms of an inequality be changed, the sign of inequality must be reversed. 201. Again, if a>b, then -a<-6, and therefore - ac<-bc; that is, if the sides of an inequality be multiplied by the same negative quantity, the sign of inequality must be reversed. 202. If a,>b, a,>b2, az>bg, ... Am>bm, it is clear that a, +az+az + ... +am>b, +62 +63 + ... +bm; and ajazdz. Am>b, babz ... bm. 203. If a>b, and if p, q are positive integers, then Va> vb, P, 1 1 or a?>b?; and therefore a?>bo; that is, an>b", where n is any positive quantity. 1 1 204. The subtraction of two inequalities subsisting in the same sense does not necessarily give an inequality subsisting in the same senise. 205. The division of an inequality by another subsisting in the same sense does not necessarily give an inequality subsisting in the same sense. P The truth of these last statements is readily seen by consid. ering the inequalities 5>4, 3>2. Subtracting member for member would give 2>2. 5 Dividing member by member would give 3>2. Example 1. Find limit of x in the inequality 5 11 5 15 Clearing of fractions, we have 15x – 25>3x +11. Transposing and combining 12x>36; .: *>3. Example 2. If a, b, c denote positive quantities, prove that a? +62+c2>bc+catab. For 62 +c2>2bc, c2+ a2>2ca, a2+12>2ab. Whence by addition a2 +62+c2>bc+ca +ab. Example 3. If u may have any real value find which is the greater, x3+1 or x2+x. 2c3+1-(x2+x) = 23 – 22 – (2 — 1) = (2-1) (-1) =(x-1)2(x+1). Now (x - 1)2 is positive, hence 23 +1> or < ? + according as x+1 is positive or negative; that is, according as a> or <-1. If x=-1, the inequality becomes an equality. EXAMPLES XXV. 46 5x 3 3 a+b 6. Show that the sum of any real positive quantity and its reciprocal is never less than 2. 7. If a + b2=1, and x2 + y2=1, show that ax+by<1. ax+by+cz<1. 2ab 2 a+b 10. Show that (22y+y+z+z2c) (xy2+yz2+z22) >9:c2y42. 11. Find which is the greater 3ab2 or a8 +268. 12. Prove that a3b+ab8<a4 +64. 13. Prove that 6abc<bc(b+c) +ca(c+a+ab(a+b). 14. Show that b2c2 +c2a2 + a2b2>abc(a+b+c). 15. Show that 2(Q8+68+c8)>bc(6+c)+ca(c+a)+ab(a+b). |