Tell the character of the above roots, whether real or imaginary : 1. If p and q are positive. 2. If p is negative and q positive. 3. If p is positive and q negative, and q<p2, merically. 4. If p is positive and q negative, and q>1-p2, merically. 6. If p is negative and q negative, and q<-p2, merically. nu nu nu 6. If p is negative and q negative, and q>1p2, p2, nu merically. Tell whether the above roots are rational or irrational : 9. If q is negative, and numerically equal to 100 10. If p = : 0. Give the signs of the above roots, and tell which root is numerically the greater: 11. If Р and q are both positive. 12. If p is negative and q positive. 13. If p is positive, q negative, and cally. 1 > q, numeri 14. If p and q are negative, and p2 > q, numerically. 4. 1 15. If p and q are negative, and p2=q, numerically. 4 What are the values of p and q in the following equations: In the following equations, are the roots 1. Real or imaginary? 2. Rational or irrational? 3. Positive or negative? 4. What are their relative values? 267. An expression denoting that two quantities are unequal in value is an Inequality. 268. The symbol of inequality is >, read greater than ; or<, read less than. 269. The quantities compared in an inequality are the members of the inequality. 270. Two inequalities are said to subsist in the same sense, when the first members are both greater or both less than the second members. 271. Two inequalities are said to subsist in an opposite or contrary sense, when the first member of the one is the greater and the second member of the other. 272. A negative quantity is considered less than a positive quantity, whatever their absolute values. 273. The process of changing the form of an inequality without changing its sense is transformation. 274. The following principles of transformation may readily be illustrated : Prin. 114.—1. The same or equal quantities may be added to both members of an inequality. 2. The same or equal quantities may be subtracted from both members of an inequality. 3. Both members of an inequality may be multiplied by the same or equal positive quantities. 4. Both members of an inequality may be divided by the same or equal positive quantities. 5. Two unequal positive members may be raised to the same power. 6. Two unequal positive members may have the same root extracted, provided the positive results only are compared. 7. The sum of two inequalities, subsisting in the same sense, may be taken member by member. 275. (a — b)2 > 0 whether a >b or b> a [P. 27]. Expanding, a2-2ab+b2 > 0 (1) Add 2 ab to both members [P. 114, 1] a2 + b2 > 2 a b. Therefore, Prin. 115.—The sum of the squares of two unequal quantities is greater than twice their product. Adding member by member [P. 114, 7], 2 a2 + 2 b2 + 2 c2 > 2 a b + 2 ac+2bc Dividing by 2 [P. 114, 4], a2 + b2 + c2 > ab+ac+bc. Therefore, (1) (2) (3) (4) Prin. 116.-The sum of the squares of three unequal quantities is greater than the sum of their products taken two and two. 2. Examples. Illustration. Which is the greater, a3 +63 or ab+ a b3, for any positive values of a and b? Solution: Dividing by (a + b), a3 + b3 > = <a2 b + a b2 (a + b) (a2 - ab + b2) > = < a b (a + b) Adding ab to both members, But a2+ b2 >2 ab [P. 115], a2 + b2 >=< 2 ab. ... a3 + b3 > a2 b + a b2, since no operation has been performed to change the sense of the inequality. EXERCISE 126. Prove the following statements true for unequal positive values of the letters: 1. a3 + a b2 > 2 a2 b 2. a3 b + a b1> a2 b2 + a b3 3. (a + b)2 > 4 ab 4. (a + b)3 > 4a2 b+4 a b2 5. a2 + 3 b2 > 2 a b + 2 b2 7. a1 + a2 b2 + a2 c2 > a3 b + a3 c + a2 b c 9. If x2+4x > 12, show that x > 2 10. If 3x+5x > 42, show that x> 3 11. If 7 x 3 x < 160, show that x <5 8. a 2a-1 14. x2 y +x y2 + x2 z + x z2 + y2 z + y z2 > 6 x y z 19. What integral value of x will satisfy 3x2+ 4 x > 64 and 3x+4x < 132 ? CHAPTER VII. RATIO, PROPORTION, AND Ratio. 1. Definitions and Principles. 277. A relation of values exists between two similar quantities-that is, one of them is a number of times or a part of the other. 278. The relation which the value of one quantity bears to that of another is the Ratio of the quantities, and is obtained by dividing the quantity compared by the quantity with which it is compared. 3 Illustration. The ratio of 3 apples to 5 apples is 5 3 since 3 apples = of 5 apples. 5 279. A ratio is expressed by writing a colon between the quantities compared, or by a common fraction. α Illustration.-The ratio of a to b is a : b, or b' 280. The quantity compared, or the first term of a ratio, is the antecedent; and the quantity with which the comparison is made, or the second term of the ratio, is the consequent. 281. Since the ratio is obtained by dividing the quantity compared by the quantity with which it is compared, it follows that, |