But if the signs of both members of an inequality are not positive, the resulting inequality may exist in the same, or in a contrary sense. Thus, 3-2, and 32>(-2)2, or 9>4. But, 3-2, and (3)>(-2)2, or 9>4. EXAMPLES INVOLVING THE PRINCIPLES OF INEQUALITIES. 1. Five times a certain whole number increased by 4, is greater than twice the number increased by 19; and 5 times the number diminished by 4, is less than 4 times the number increased by 4. Required the number. 5x-2x19-4, from eq. (1) by transposing, 3x15, by reducing, x>5, by dividing both members by 3. 5x-4x 4+4, from eq. (2) by transposing, x8, by reducing. Hence, the number is greater than 5 and less than 8, consequently either 6 or 7 will fulfill the conditions. 2. If 4x-7-2x+3, and 3x+1>13-x, find x. Ans. x=4. 3. Find the limit of x in 7x-3>32. Ans. x>5. 4. Of x in the inequality 5+18+1x Ans. <36. bc cd cf (Art. 221.) 6. It is required to prove that the sum of the squares of any two unequal magnitudes is always greater than twice their product. Since the square of every quantity, whether positive or negative, is positive, it follows that (a—b)2, or a2-2ab+b2>0. Adding, +2ab to each side (Art. 219), a2+b2>2ab, which was required to be proved. Most of the inequalities usually met with, are made to depend ultimately upon this principle. 7. Which is greater, √/5+√/14 or √/3+3√2? Ans. the former. 8. Given (x+2)+}x<¦(x−4)+3 and >¦(x+1)+}, to find x. Ans. x=5. 9. The double of a certain number increased by 7, is not greater than 19, and its triple diminished by 5, is not less than 13. Required the number. Ans. 6. 10. Show that every fraction the fraction inverted, a b is greater than 2; that is, that + 2. b a 11. Show that a2+b2+c2>ab+ac+bc, unless a=b=c. 12. If x2=a2+b2, and y2=c2+d2, which is greater, y or ac+bd? Ans. xy. 13. Show that abc>(a+b—c)(a+c—b)(b+c—a), unless abc. VII. QUADRATIC EQUATIONS. 224. A Quadratic Equation, or an equation of the second degree, is one in which the greatest exponent of the unknown quantity is 2; as, x2+x=a. An equation containing two or more unknown quantities, in which the greatest sum of the exponents of the unknown quantities in one term is 2, is also a Quadratic Equation; as, xy=a, xy-x-y=c. 225. Quadratic equations, containing only one unknown quantity, are divided into two classes, pure and affected. A Pure Quadratic Equation is one that contains only the second power of the unknown quantity, and known terms; as, x2+2=47—4x2, and ax2+b=cx2—d. A pure quadratic equation is also called an incomplete equation of the second degree. An Affected Quadratic Equation is one that contains both the first and second power of the unknown quantity, and known terms; as, 5x2+7x=34, and ax2-bx2+cx-dx-e-f. An affected quadratic equation is also called a complete equation of the second degree. 226. The general form of a pure equation is ax2=b. The general form of an affected equation is ax2+bx=c. Every quadratic equation containing only one unknown. quantity may be reduced to one of these forms. For, in a pure equation, all the terms containing x2 may be collected into one term of the form, ax2; and all the known quantities into another, as b. So, in an affected equation, all the terms containing may be reduced to one term, as ax2; and those containing x to one, as bx; and the known terms to one, as c. PURE QUADRATIC EQUATIONS. 227.-1. Let it be required to find the value of x in 2—3+51⁄2x2=123—x2. the equation, Clearing of fractions, 4x2-36+5x2-153-12x2; Since the square of -3 is the same as the square of +3, the -3, will give the same result as x+3. value x 2. Given ax2+b=d+cx2, to find the value of x. Rule for the Solution of a Pure Equation.-Reduce the equation to the form ax2-b. Divide by the coefficient of x, and extract the square root of both members. 228. If we solve the equation ax2=b, we have, The equation may be verified by substituting either of these values of x. Hence, we infer, 1st. That in every pure equation the unknown quantity has two values, or roots, and only two. 2d. That these roots are equal in value, but have contrary signs. 1. 11x-44-5x+10. 2. }(x2—12)=¦¤2 ́1.. 3. (x+2)2=4x+5.. 229. For the statement of the equation, see Art. 154. 1. What two numbers have the ratio of 2 to 5, the sum of whose squares is 261? Let 2x and 5x the numbers. Hence, 2x=6, and 5x-15 the required numbers. |