208. An inequality will continue to subsist in the same sense if each member is increased, diminished, multiplied, or divided, by the same positive number. Thus, if a>b, then a + c > b + c; a-c>b-c; ac > bc; a÷c>b÷c. Therefore, 209. A term can be transposed from one member of an inequality to the other, provided the sign of the term is changed. Thus, if a-cb, by adding c to both members, a > b + c. (§ 208) 210. An inequality will be reversed if its members are subtracted from equal numbers; or if its members are multiplied or divided by the same negative number. Thus, if x = y and a>b, then x - a <y-b; - ac <-bc; and a ÷ (- c) <b÷ (- c). 211. The sum or product of the corresponding members of two inequalities that subsist in the same sense is an inequality in the same sense. Thus, if a band c>d, then a + c > b + d, and ac bd. 212. The difference or quotient of the corresponding members of two inequalities that subsist in the same sense may be an inequality in the same sense, or the reverse, or may be an equality. 8. Twice a certain integral number increased by 7 is not greater than 19; and three times the number diminished by 5 is not less than 13. Find the number. 9. Twice the number of pupils in a certain class is less than 3 times the number minus 39; and 4 times the number plus 20 is greater than 5 times the number minus 21. Find the number of pupils in the class. 213. Theorem. If a and b are unequal, a2 + b2 > 2 ab. For (a - b)2 must be positive, whatever the values of a and b. If a and b are positive and unequal, show that a2 + b2 > 2 ab. If the letters are unequal and positive, show that: 1. a2 + 3b2 > 26 (a + b). 2. (a2 + b2) (a2 + b2) > (a3+b3)2. 3. a2b+ a2c + ab2 + b2c + ac2 + bc2 > 6 abc. 4. The sum of any fraction and its reciprocal > 2. 5. ab + ac+bc <(a+b - c)2 + (a+c-b)2+(b+c-a)2. 6. (a2 + b2) (c2 + d2) > (ac+bd)2. CHAPTER XV. INVOLUTION AND EVOLUTION. Involution. 214. The operation of raising an expression to any required power is called Involution. 215. Index Law for Involution. If mis a positive integer, am = axaxa............to m factors. Consequently, if m and n are both positive integers, (an)m = a × a" × a" ..... to m factors Any required power of a given power of a number is found by multiplying the exponent of the given power by the exроnent of the required power. = (a × a ............ to n factors) (b × b ..... to n factors) = anbn. In like manner, (abc)" = a"b"en; and so on. Hence, Any required power of a product is found by taking the product of its factors each raised to the required power. 217. In the same way it may be shown that Any required power of a fraction is found by taking the required power of the numerator and of the denominator. 218. From the Law of Signs in multiplication it is evident that all even powers of a number are positive; all odd powers of a number have the same sign as the number itself. Hence, no even power of any number can be negative; and the even powers of two compound expressions that have the same terms with opposite signs are identical. Thus, 2 (b - a)2= (a - b)2. 219. Binomials. By actual multiplication we obtain, (a + b)2 = a2+2ab+b2; (a + b)3 = a3+3a2b+3ab2 + b2; (a + b)2 = a2 + 4 ab + 6 a2b2 + 4 ab3 + 64. In these results it will be observed that: 1. The number of terms is greater by one than the exponent of the binomial. 2. In the first term the exponent of a is the same as the exponent of the binomial, and the exponent of a decreases by one in each succeeding term. 3. b appears in the second term with 1 for an exponent, and its exponent increases by 1 in each succeeding term. 4. The coefficient of the first term is 1. 5. The coefficient of the second term is the same as the exponent of the binomial. 6. The coefficient of each succeeding term is found from the next preceding term by multiplying the coefficient of that term by the exponent of a and dividing the product by a number greater by one than the exponent of b. 220. If b is negative, the terms in which the odd powers of b occur are negative. Thus, |