CHAPTER XXIII INEQUALITIES* 383. An inequality is a statement that one quantity is greater or less than another. 384. The signs of inequality, > and <, are respectively read "is greater than" and "is less than." The members or sides of an inequality are the two expressions which are connected by a sign of inequality. Thus, ab is read "a is greater than b." a is the left, b the right, member. 385. One number a is greater than another number b, if a-b is positive; similarly, a < b, if a − b is negative. - 5-7, since - 5-(-7) = +2. 386. The symbols, >, <, ‡, express respectively "is not greater than," "is not smaller than," and "is not equal to." 387. Two inequalities are of the same species, or subsist in the same sense, if their signs of inequality are alike. a>b, c>d, are of the same species. 4>3, -4<-3, are of opposite species. 388. The sense of an inequality is not changed if both members are increased or diminished by the same number. Suppose ab, then ab is positive. * This chapter is not required for the examinations of the College Entrance Examination Board. 389. It follows from the preceding paragraph that a term may be transposed from one member of an inequality to another, by changing its sign. 390. The sense of an inequality is not changed if both members are multiplied or divided by the same positive number. Let a >b, and m be a positive number. Then ab is positive. Hence m(a - b) or ma mb must be positive; i.e. ma>mb. In a similar manner it can be proved that > a b m m 391. If the signs of all terms of an inequality are changed, the sign of inequality must be reversed. Consider the inequality a-b+c>x-y. Transposing all terms, That is, -x+ya+b-c. a+b-c<-x+y. 392. The sense of an inequality is reversed if both members are multiplied or divided by the same negative number. This follows from § 390 and § 391. 393. The following principles can easily be demonstrated : 1. If any number of inequalities of the same species be added, the resulting inequality will be of the same species as the original ones. 2. If a>b, and b>c, then a > c. 3. The sense of inequalities of the same species is not changed by multiplying their corresponding members, provided all members are positive. HINT. If a >b, and x>y, then ax> bx, and bx>by. Therefore ax > by. 4. The sense of an inequality is not changed by raising both numbers to the same power, if the signs of the members are not changed by the involution. 394. An identical inequality is one which is true for all values of the letters involved. Thus, a +9a, and m2 + n2>0, are identical inequalities. 395. A conditional inequality is one which is true only for certain values of the letters involved. z+15 is a conditional inequality. It is true only if z>4. 396. To solve a conditional inequality for a certain letter means to find all values of the letter for which the inequality is true. Thus, z-1>2 is evidently true for all values of z greater than 3. The value 3 is sometimes called the limit of z" in the inequality z −1>2. 397. Conditional inequalities are solved in the same manner as equations. Care, however, should be taken to change the sense of an inequality if the signs of both members are changed by multiplication, division, etc. Ex. 1. Solve the inequality: 6–5x_2x+3>1-x-5 12 6 Clearing of fractions, 6-5x-4x-6 > 12 - 4x + 20. 398. Identical inequalities are usually proved by the same method as identical equations (§ 169). The fact that the square of any real number is positive can often be used for such proofs. a Ex. 2. If a and b are unequal, positive numbers, prove that NOTE. All letters in the present chapter are supposed to represent real numbers. 399. If a=b, then a2 + b2 = 2 ab; if ab, then a2 + b2>2 ab. Hence a2+b2 is either equal to or greater than 2 ab, a statement that may be written: a2 + b2 ≥2 ab, a2+b2 2.ab. (1) (2) or, Many proofs of inequalities can be based upon the iden tity (1). Ex. 3. Prove that a2+b+c2 ≥ ab+bc + ac. |