174. Hence we conclude that we may transpose a term from one member of an inequality to the other, provided we change its sign. Thus, suppose a2+b2>3b2-2a2. Adding 2a2 to each member of the inequality, it becomes a2+b2+2a2>3b2. Subtracting b2 from each member, we have 175. 2d. If we add together the corresponding members of two or more inequalities which subsist in the same sense, the resulting inequality will always subsist in the same sense. 176. 3d. If one inequality be subtracted from another which subsists in the same sense, the result will not always be an inequality subsisting in the same sense. Take the two inequalities 4<7 Subtracting, we have 4-2<7-3, or 2<4, where the result is an inequality subsisting in the same sense. But take and 9<10 Subtracting, we have 9-6>10-8, or 3>2, where the result is an inequality subsisting in the contrary sense. We should therefore avoid as much as possible the use of this transformation, or, when we employ it, determine in what sense the resulting inequality subsists. 177. 4th. If we multiply or divide each member of an inequality by the same positive quantity, the resulting inequality will subsist in the same sense. Hence an inequality may be cleared of fractions. Thus, suppose we have a2-b2 c2-d2 2d За Multiplying each member by 6ad, it becomes 3a (a2-b2)>2d(c2—d2). 178. 5th. If we multiply or divide each member of an inequality by the same negative number, the resulting inequality will subsist in the contrary sense. Multiplying each member by -3, we have the opposite in equality So, also, -24<-21. 15>12. Dividing each member by -3, we have -5 -4. Therefore, if we multiply or divide the two members of an inequality by an algebraic quantity, it is necessary to ascertain whether the multiplier or divisor is negative, for in this case the resulting inequality subsists in a contrary sense. 179. 6th. If the signs of all the terms of an inequality be changed, the sign of inequality must be reversed. For to change all the signs is equivalent to multiplying each member of the inequality by -1. 180. Reduction of Inequalities.—The principles now established enable us to reduce an inequality so that the unknown quantity may stand alone as one member of the inequality. The other member will then denote one limit of the unknown 7. A man, being asked how many dollars he gave for his watch, replied, If you multiply the price by 4, and to the product add 60, the sum will exceed 256; but if you multiply the price by 3, and from the product subtract 40, the remainder will be less than 113. Required the price of the watch. 8. What number is that whose half and third part added together are less than 105; but its half diminished by its fifth part is greater than 33? 9. The double of a number diminished by 6 is greater than 22, and triple the number diminished by 6 is less than double the number increased by 10. Required the number. CHAPTER XI. INVOLUTION. 181. A power of a quantity is the product obtained by tak ing that quantity any number of times as a factor. Thus the first power of 3 is 3; the second power of 3 is 3x3, or 9; the fourth power of 3 is 3×3×3×3, or 81, etc. Involution is the process of raising a quantity to any power. 182. A power is indicated by means of an exponent. The exponent is a number or letter written a little above a quantity to the right, and shows how many times that quantity is taken as a factor. Thus the first power of a is a1, where the exponent is 1, which, however, is commonly omitted. The second power of a is a xa, or a2, where the exponent 2 denotes that a is taken twice as a factor to produce the power аа. The third power of a is a xaxa, or a3, where the exponent 3 denotes that a is taken three times as a factor to produce the power aaa. The fourth power of a is a×a×a×a, or aa. Also the nth power of a is a×a×a×a, etc., or a repeated as a factor n times, and is written a". The second power is commonly called the square, and the third power the cube. 183. Exponents may be applied to polynomials as well as to monomials. Thus (a+b+c)3 is the same as (a+b+c)×(a+b+c)×(a+b+c), or the third power of the entire expression a+b+c. Powers of Monomials. 184. Let it be required to find the third power or cube of 2a3b2. According to the rule for multiplication, we have (2a3b2)3=2a3b2 × 2a3b2 × 2a3b2=2×2×2a3a3a3b2b2b2=8a°b*. In a similar manner any monomial may be raised to any power. Hence, to raise a monomial to any power, we have the following RULE. Raise the numerical coefficient to the required power, and multiply the exponent of each of the letters by the exponent of the required power. 185. Sign of the Power.-With respect to the signs, it is obvious from the rules for multiplication that if the given monomial be positive, all of its powers are positive; but if the monomial be negative, its square is positive, its cube negative, its fourth power positive, and so on. Thus — a ×—a=+a2, In general, any even power of a negative quantity is positive, and every odd power negative; but all powers of a positive quantity are positive. EXAMPLES. 1. Find the square of 11a2bcd2. 2. Find the square of -18x2yz3. 3. Find the cube of 7ab2x2. 4. Find the cube of -8xy2z3. 5. Find the fourth power of 4ab2c3. Ans. 121ab2c2d4. |