CHAPTER XIV. INEQUALITIES. 201. If ab is positive, a is said to be greater than b; if a b is negative, a is said to be less than b. NOTE. Letters in this chapter are understood to stand for positive numbers, unless the contrary is expressly stated. 202. An Inequality is a statement in symbols that one of two numbers is greater than or less than the other. 203. The Sign of an Inequality is >, which always points toward the smaller number. Thus, ab is read a is greater than b; c <d is read c is less than d. 204. The expressions that precede and follow the sign of an inequality are called, respectively, the first and second members of the inequality. 205. Two inequalities are said to subsist in the same sense if the signs of the inequalities point in the same direction; and two inequalities are said to be the reverse of each other if the signs point in opposite directions. Thus, ab and c>d subsist in the same sense, but ab and c<d are the reverse of each other. 206. If the signs of all the terms of an inequality are changed, the inequality is reversed. Thus, if a>b, then — a <- b. 207. If the members of an inequality are interchanged, the inequality is reversed. Thus, if a >b, then b<a. 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-c>b, 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 ab, then x-ay-b; a ÷ (c)<b÷ (−c). ac<-bc; and 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 ab and 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. That is, (a - b)2 > 0. a2-2ab+b2 >0. Squaring, If a and b are positive and unequal, show that If the letters are unequal and positive, show that: 1. a2 + 3 b2 > 2b (a + b). 2. (a2 + b2) (a1 +b1) > (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)? 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. am = ax a x a ..... If m is a positive integer, to m factors. Consequently, if m and n are both positive integers, Any required power of a given power of a number is found by multiplying the exponent of the given power by the exponent of the required power. 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. |