16. Find the 10th term only of (x − 4√a)1. 17. Find the term containing xin (x - 4√a). 18. Find the term containing " in (√xVxy)11. 19. Expand (1 + 2x − x2)4. 10 CHAPTER XXI INEQUALITIES 161. Problems presented so far have been solved by statements of equality between certain numbers, but some problems may contain conditions that lead to statements that one number is greater or less than another. It is the purpose of this chapter to show how such statements may yield solutions of the problems. An inequality is an algebraic statement that one number is greater or less than another. One number is greater than another when the remainder obtained by subtracting the second from the first is positive, and is less than another when the remainder obtained by subtracting the second from the first is negative. Thus, if a - b is a positive number, a is greater than b; but if a - b is a negative number, a is less than b. 2 In the scale of numbers, . . . +4, +3, +2, +1, 0, −1, -3, ..., any negative number is less than 0 and of two negative numbers, the one farther away from 0 is the less. Thus, -1 is less than 0 and −4 is less than −3. The sign of inequality is> or <. It is placed between two unequal numbers always pointing towards the smaller. Thus a > b, is read, a is greater than b and a < b is read, a is less than b. (In this chapter all letters represent real positive numbers unless otherwise specified.) The expression on the left of the sign of inequality is called the first member, and that on the right, the second member of the inequality. The signs and are sometimes used, and are read respectively, is not greater than and is not less than. If two inequalities have their signs pointing in the same direction they are said to subsist in the same sense. As, a > b and c > d. If the signs point in opposite directions, the two inequalities are said to subsist in a contrary sense. As, x>y and m < n. 162. In many respects, an inequality may be treated like an equation. Principle 1. If the same number or equal numbers be added to or subtracted from both members of an inequality, the resulting inequality will subsist in the same sense. Principle 2.-If both members of an inequality are multiplied or divided by the same number, the resulting inequality will subsist in the same sense. As the transposition of a term is equivalent to adding or subtracting the same number, hence Principle 3.-A term may be transposed from one side of the inequality to the other, provided its sign is changed. Principle 4.—If the corresponding members of any number of inequalities subsisting in the same sense are added, the resulting inequality will subsist in the same sense. If 5 > 3 Multiply, 15 > 6 Hence, Principle 5. If the corresponding members of two inequalities subsisting in the same sense are multiplied to gether, the resulting inequality will subsist in the same sense, provided all the members are positive. Since +5 +3 but −5 < −3. Principle 6.-If the signs of all the terms of an inequality are changed, the resulting inequality will subsist in a contrary · sense. Hence, any operation that changes the signs of the terms will reverse the sign of inequality. Principle 7.-If both members of an inequality are multiplied or divided by the same negative number, the resulting inequality will subsist in a contrary sense. If two unequal numbers are subtracted from two equal numbers, the subtraction of the greater number will leave a smaller number. Thus, 10 = 10 Subtract 5 > 3 5 < 7 Also 10 = 10 Subtract -5 < −3 15 > 13 Principle 8.—If each member of an inequality is subtracted from the corresponding member of an equation, the resulting inequality will subsist in a contrary sense. In the following operations, the result may have one member greater than, equal to, or less than the second member. Sum of two inequalities subsisting in a contrary sense. Subtract 7 > 3 Subtract 10 > 3 Subtract 10 > 2 7 > 4 4 = 4 4 < 5 Product of two inequalities when some of the members are negative. Divide by 2 > 1 Divide by 7> 3 Divide by 7 > 2 That is, when x has any value greater than 13 the inequality is satisfied. 1 is called the inferior limit of x. < x < 7 Therefore the values of x lie between 3 and 7. The inferior limit of x is 3, and the superior limit is 7. 4. 4x6 7x - 1. 5. 9x2 3x + 10. 4x - 2 >7 5 |