-48-105. Also, from -2>-3 we get (-2)2 <(— 3)2 7. If we take A>-B or BA, and suppose A to be positive, while B is negative, then we shall have A2 > B2 or B2 < A2 if B is numerically less than A, and the reverse when B is numerically greater than A. Also, from A> -B we clearly have A3> - B3 or — B3 < A3. Hence, if the members of an inequality have unlike signs, and are raised to a power whose index is an even positive integer, the powers will form an inequality which will exist in the same sense as the given inequality, if the negative member is numerically less than the positive member, and vice versa if the negative member is numerically greater than the positive member; and the reverse will be true in each case if the index of the power is taken with the negative sign. Also, if the members of such an inequality are raised to a power whose index is an odd positive or negative integer, the result will be an inequality which will exist in the same sense as the given inequality. Thus, from 3-2 and 4 > - 5 we get (3) > (-2)2 and (4)2 < (— 5)2 or 9> 4 and 16 <25; and (3)-° < (− 2)−3, (4)-2 > (— 5)-2, or < 1 1 1 1 16 25 Also, from 3>-2 we get (3)3 > (— 2)3 and (3)-3 > (-2)-3, or 27 > - 8 and 8. From A > B or B < A we get A> B" when ʼn is an odd positive integer, or A- <B-; and the same results hold good if n is an even integer, when the positive roots are used; but if we take the negative roots (n being even), we — A* < – B" and — A− > – B-. shall get Also, if we have A > B and A <- B; then if n is an odd positive integer, we shall have (A)> (— B) and (— A)" < (— B)* and (— A)−✯ > (— B)−¦. Thus, from 27 > 8 we get (27)3 > (8)3 and (27)−¦ < (8)−↓ or 3 > 2 and 1 1 < and from 9> 4 there results (9) > (4) 3 2 <;; also, − (9)3 < − (4) 3 (9)− § < (4)−1, or 3>2 and 1<<1 3 and — (9)− ¦ > — (4)—§, or — 3 < — 2 and − 1 > — 1; — — 3 2 ; and from 8 - 27 and 8>- 27 we get (8)* > (→ 27)+} or -- (— 2±1 > (— 3)±1, and (— 8)3 > (— 27)3, (— 8)−3 < (— 27)-}, ine Hence, we can extract any root of the members of any quality when the index of the root is an odd positive integer, and the result will be an inequality which will exist in the same sense as the given inequality; and if the members of the given inequality are both positive, and we extract any even positive root of its members, the same conclusions hold true when the positive roots are used; but if the positive root is taken in one member and the negative in the other, the positive root will be greater than the negative root. - 9. If we have AB+ C, and suppose B and C to be positive, then by rejecting C we have the inequality A > B or BA; and if we have A- B+C, B being negative and C positive, then by rejecting C, we get the inequality A> B. Hence, we perceive how inequalities may be conceived to result from equations. 10. To illustrate what has been done, take the following EXAMPLES. 1. Given, 2x-5 1, to find the limit of x. Transposing 5 and dividing by 2, we have > 3; conse quently, since a must be greater than 3, we may call 3 the inferior limit of x. 2. Given, 2x-34, to find the limit of x. Proceeding as before, we get a <31; consequently, since a must be less than we may call 31 the superior limit 2' 2 of x. 3. Given, 2x-3 > 1 and 3x + 1 <16, to find the limits of x. From the first inequality we get a > 2, and from the second we have a <5; consequently, 2 and 5 are the inferior and superior limits of x. 1 Freeing the inequalities from fractions, etc., we get x> 7 and <1. 5. Given, 2-3 > 1 and 23 + 2 <29, to find the limits of x. The inequalities are equivalent to a2 > 4 and 3 <27; consequently, by extracting the roots we get a > 2 and ≈ <3. 6. Given, 9 and limits of x. -a64, to find the By changing the signs of the inequalities, we get x2 > 9 and 64; consequently, from the extraction of the roots, we have > 3 and <4. 7. Given, 3x + 2y > 7 — 2x + y and 5x-7y < 2, to find the limits of x and y. The inequalities give > 7 — Y and x < Ty + 2; conse 5 5 quently, we must have 7y+2>7-y, which gives y> 8 8. A person's estate being increased by its third part and then diminished by $4000 is greater than $3900; also, the estate being diminished by its fourth part and then by $4400 is less than $150. What are the limits of the estate. Ans. $5925 and $60663. 9. A person sells two valuable horses for a sum which is not less than $600, and the difference of the prices of the horses is not greater than $80. What are the limits of the money received for each horse? Ans. Not less than $340 for the most valuable, and not less than $260 for the other. SECTION XIV. QUADRATIC EQUATIONS. (1.) THERE are two kinds of quadratic equations. 1st. Those which involve only the square of the unknown letter, which are called incomplete, simple, or pure quadratics, such as 29, 5x2 -7 + x2 = − x2 + 21, 2 3 and a+b= c. They are sometimes called x2 = 8+ 6' quadratic equations of two terms, because, by transposition, etc., they can always be reduced to the form aa2=b, or a = b a Thus, 57+x2-a +21 by transposition be = comes 5x2 + x2 + x2 = 21 + 7, and uniting like terms, and 5x2 dividing by 7, we have a 4. Similarly, the equation 2 = -5=8+ when freed from fractions, is easily re x2 extract the square root, both in the arithmetical and alge tion x2 = b a b a a α an identical equation; consequently, the equa has two roots, which are expressed by ±√, or, which is the same, the equation admits of two solutions, one of which is obtained by putting putting-√ for a. a b for x, and the other by a = 2d. Those equations which involve the square and first power of the unknown letter-which are called adfected, affected, or complete quadratics-such as 32-43 18 x2 8x + =40, and ax2+bx=c. They are some3 5 3 4 -2x-x2, times called quadratic equations of three terms, because, by transposition, etc., they can always be reduced to the form ax2 + bx=c, or x2 + x = -· Thus 3x-4x-318-2x-x2 b a с α by transposition becomes 3+x2-4x+2x=18+ 3; and uniting like terms, we have 4a2-2x= 21, or dividing by the coefficient of x2, we get a2 — and reduced as before, 24x 5 Extracting the square root of 2+ by the common |