cannot be satisfied by any finite value of x. We will however remark b that, as the equation can be put under the form = 0, if we sub stitute for x, numbers greater and greater, will differ less and less from 0, and the equation will become more and more exact; so b that, we may take a value for a so great that will be less than It is in consequence of this that algebraists say, that infinity satisfies the equation in this case; and there are some questions for which this kind of result forms a true solution; at least, it is certain that the equation does not admit of a solution in finite numbers, and this is all that we wish to prove. 2d. If we have a=0, b=0, at the same time, the value of x In this case, the equation becomes 0xx=0, and every finite number, positive or negative, substituted for x, will satisfy the equation. Therefore the equation, or the problem of which it is the algebraic translation, is indeterminate. does not al. 113. It should be observed, that the expression ways indicate an indetermination, it frequently indicates only the existence of a common factor to the two terms of the fraction, which factor becomes nothing, in consequence of a particular hypothesis. x= For example, suppose that we find for the solution of a problem, a3-b3 a2-b2 0 If, in this formula, a is made equal to b, there results =이 But it will be observed, that a3-b3 can be put under the form (a−b) (a2+ab+b2), (Art. 59), and that a2-b2 is equal to (a-b) (a+b), therefore the value of x becomes (a−b) (a2+ab+b2) x= (a-b) (a+b) Now, if we suppress the common factor (a-b), before making the supposition a=b, the value of a becomes x= which reduces to x= 3a2 За or x=2, when a=b. 2a. 2' For another example, take the expression a2-b2 (a+b) (a−b) (a--b)(a−b) (a−b) Making a=b, we find 2 0 x=이 a2+ab+b2 a+b because the factor (a-b) is common to the two terms; but if we first suppress this factor, there re sults x= a+b , 2a which reduces to x= when a=b. 0 From this we conclude, that the symbol 이 sometimes indicates the existence of a common factor to the two terms of the fraction which reduces to this form. Therefore, before pronouncing upon the true value of the fraction, it is necessary to ascertain whether the two terms do not contain a common factor. If they do not, we conclude that the equation is really indeterminate. If they do contain one, suppress it, and then make the particular hypothesis; this will give the true value of the fraction, which will assume one of the three forms AAO 명인인 in which case, the equation is determi nate, impossible in finite numbers, or indeterminate. This observation is very useful in the discussion of problems. Of Inequalities. 114. In the discussion of problems, we have often occasion to suppose several inequalities, and to perform transformations upon them, analogous to those executed upon equalities. We are often obliged to do this, when, in discussing a problem, we wish to establish the necessary relations between the given quantities, in order that the problem may be susceptible of a direct, or at least a real solution, and to fix, with the aid of these relations, the limits between which the particular values of certain given quantities must be found, in order that the enunciation may fulfil a particular condition. Now, although the principles established for equations are in general applicable to inequalities, there are nevertheless some exceptions, of which it is necessary to speak, in order to put the beginner upon his guard against some errors that he might commit, in making use of the sign of inequality. These exceptions arise from the introduction of negative expressions into the calculus, as quantities. In order that we may be clearly understood, we will take exam. ples of the different transformations that inequalities may be subjected to, taking care to point out the exceptions to which these transformations are liable. 115. Two inequalities are said to subsist in the same sense, when the greater quantity stands at the left in both, or at the right in both; and in a contrary sense, when the greater quantity stands at the right in one, and at the left in the other. Thus, 25>20 and 18>10, or 6<8 and 7<9, are inequalities which subsist in the same sense ; and the inequalities 15>13 and 12<14, subsist in a contrary sense. 1. If we add the same quantity to both members of an inequality, or subtract the same quantity from both members, the resulting inequality will subsist in the same sense. Thus, take 8>6; by adding 5, we still have 8+5>6+5 and 8-56-5. When the two members of an equality are both negative, that one is the least, algebraically considered, which contains the greatest number of units. Thus, -25<-20; and if 30 be added to both members, we have 5<10. This must be understood entirely in an algebraic sense, and arises from the convention before established, to consider all quantities preceded by the minus sign, as subtractive. The principle first enunciated, serves to transpose certain terms from one member of the inequality to the other. Take, for example, the inequality a2+b2>3b-2a2; there will result from it a2+2a2>362-b2, or 3a2>262. 2. If two inequalities subsist in the same sense, and we add them member to member, the resulting inequality will also subsist in the same sense. Thus, from a>b, c>d, e>f, there results a+c+e>b+d+f. But this is not always the case, when we subtract, member from member, two inequalities established in the same sense. Let there be the two inequalities 4<7 and 2<3, we have 4-2 or 2<7-3 or 4. But if we have the inequalities 9<10 and 6<8, by subtracting we have 9-6 or 3>10-8 or 2. We should then avoid this transformation as much as possible, or if we employ it, determine in what sense the resulting inequality exists. 3. If the two members of an inequality be multiplied by a positive number, the resulting inequality will exist in the same sense. Thus, from a<b, we deduce 3a<3b; and from -3a<-3b. a<-b. 3a(a2-b2)>2d(c2-d2). The same principle is true for division. But when the two members of an inequality are multiplied or divided by a negative number, the inequality subsists in a contrary sense. Take, for example, 8>7; multiplying by -3, we have -24<-21. 8 8 3 7 3 In like manner, 8>7 gives - 3, or -에이. Therefore, when the two members of an inequality are multiplied or divided by a number expressed algebraically, it is necessary to ascertain whether the multiplier or divisor is negative; for, in that case, the inequality would exist in a contrary sense. 4. It is not permitted to change the signs of the two members of an inequality unless we establish the resulting inequality in a contrary sense; for this transformation is evidently the same as multiplying the two members by -1. 5. Both members of an inequality between positive numbers can be squared, and the inequality will exist in the same sense. Thus from 5>3, we deduce 25>9; from a+b>c, we find (a+b)2>c2. 6. When both members of the inequality are not positive, we cannot tell before the operation is performed, in which sense the resulting in. equality will exist. For example, -2<3 gives (-2)2 or 4<9; but -3>-5 gives, on the contrary, (-3)2 or 9<(-5)2 or 25. We must then, before squaring, ascertain whether the two mem. bers can be considered as positive numbers. EXAMPLES. 1. Find the limit of the value of x in the expression 2. Find the limit of the value of x in the expression 3. Find the limit of the value of x in the expression 1 6 1 х 13 17 3+2+22 Ans. x>6. 4. Find the limit of the value of x in the inequalities |