. . . . . Then (1) x2 +y2 = (41)2 and (2) &xy) = 180. . . . Hence the number of ways = = =T 10.9 = 45. . . . . . . The following indicate the meaning of inequality signs. . . SOLUTION Amount from principal P in 1 year at rate r = P(l + r). . \Schaum’s Outline of Linear Algebra", S. Lipschutz and M. Lipson, McGraw-Hill 2008. . Not Enough Time? 24.4 APPLICATIONS OF LOGARITHMS The loudness, L , of a sound (in decibels) perceived by the human ear depends on the ratio of the intensity, I, of the sound to the threshold, 10, of hearing for the average human ear. . + Another method. . . 11.1 11.2 11.3 11.4 11.5 Chapter 12 58 58 58 58 59 60 SIMPLE OPERATIONS WITH COMPLEX NUMBERS . . If a person invests $3000 at 6% per year for five years, how much will the investment be worth at the end of the five years? . . - - 25.7 In how many different orders may 5 persons be seated in a row? an-2 2 an-'x + x (n - l)! . . 3 8 11 $+f=12fz=z ( 5 ) The product of two fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the fractions. . The amounts are equal if P(l + r) = P(1.03)2, 1 + r = (1.03)*, r = 0.0609 or 6.09%. . . . . (4 x + 2 y s 2 0 (b) 3x+y24, and 3x+lOys80 x+yr2, -x+yS4, and xs5 19.36 Use linear programming to solve each problem. Download pdf × Close Log In. The foci are (c,O), and (-c,O), so F ( f i / 9 , 0) and F ( - f i / 9 , 0 ) . . . . . 25.53 With six signal flags of different colors, how many different signals can be made by displaying three flags one above the other? 4 - 3 . . &, 32.5 2 - 4x - 15 ( x +2)3 * Let y = x + 2 ; then x = y - 2 . . . 2 - 1 3 2 3 1 2 4 1 -3 -1 3 -1 2 -2 -3 3 - 1 2 1 4 2 0 - 3 -2 1 -3 2 1 3 - 1 4 - (d) 3 2 - 1 3 2 -2 0 3 4 3 1 -3 -2 1 0 2 4 1 0 1 -1 -1 2 1 0 29.39 Factor each determinant: (a) I f2 a3 :2 (6) b3 c3 1 1 1 1 1 x 1 1 y 2 x2 y2 z2 x3 y3 z3 29.40 Solve each system: x - 2y + z - 3 w = 4 2 x + y - 3 z = -5 3y + 42 + w = 5 22 - w - 4x = 0 w+3x-y=4 2 x + 3 y - z - 2 w = -4 3x-4y+22-4w = 12 2 x - y - 3 z + 2 w = -2 29.41 Find il and i4 for the system I 2il - 3i3 - i4 = -4 3il + i2 - 2i3 2i4 + 24 = 0 -il - 3i2 + 2i4 + 3is = 2 il 2i3 - is = 9 2i1+i2=5 + + 1 2 - 1 1 -2 3 2 -1 3 -1 1 -4 -1 4 -3 2 DETERMINANTS OF ORDER n 352 29.42 Determine whether each system is consistent. The = causes t h e expression t o be evaluated. SOLUTION Step I. . . . . 31.9 1 + 3 + 5 + . 208 19.35 INEQUALITIES [CHAP. The formula is true for n = 1. 360 MATRICES 30.4 Write each matrix in reduced row-echelon form. . . 24.23 If $500 is borrowed for one month and $525 must be paid back at the end of the month, what is the simple annual interest? But the theorem holds for n = 1; hence it holds for n = 1 1 = 2. . . Substituting + U, y = U - U in (1) and ( 2 ) , we obtain x =U + u2 + 2u = 16 and (4) u2 - u2 + U = 11. we get 2u2 + 3u - 27 = 0 , (U - 3)(2u + 9 ) = 0 and U = 3, -9/2. SOLUTION pH = -log[H+] pH = -l0g(3.98 X 10-8) pH = -10g3.98 - (-8)log 10 pH = -0.5999 pH = 7.4001 pH = 7.40 +8 . . . Missed Lectures? . Hence 32.7 7 2 - 2 5 +~6 x-5 (x2-2x-1)(3x-2)=~-2r-1 4x2 - 28 4x2 - 28 x4+XZ-6-(X2+3)(X2-2) Ax+B =-+X2+3 4 +-3x-2' Cx+D 2-2 SOLUTION 4x2 - 28 = (Ax + B ) ( 2 - 2) + (CX+ D)(x2 + 3 ) = ( A 2 + Bx2 - ~ A -x2B) + (Cx3+ Dx2 + ~ C +X30) = (A + C1x3 + ( B + D,X2 + (3C- 2Alx - 2B + 3 0 -2B - C = 6. . Thus, the partial fraction decomposition is: 3x2+3x+7 (x - 2)2(x2 1) + -1 =-+-+x -2 5 (x - 2)2 X x2 +1 Solved Problems 32.1 Resolve into partial fractions x+2 or 2x2 - 7~- 15 x+2 (2x + 3)(x - 5 ) * PARTIAL FRACTIONS CHAP 321 SOLUTION x+2 -- A (2x + 3)(x - 5 ) - 2x + 3 Let +-- B -5 - x + - + + 375 + + A(x 5 ) B(2x 3) - ( A 2B)x 3B - 5A * (2 3)(x 5 ) (2x 3)(x - 5 ) + - We must find the constants A and B such that + + - + - ( A 2B)x 3B - SA x+2 identically (2x 3)(x 5 ) (2x 3)(x 5 ) x + 2 = ( A +2B)x + 3 B - 5A. . . . PDF. . . . Solution Rather than do all t h e computations by hand use t h e Studyworks function capabilities by assigning t h e formula t o f(x). . (U) 19.33 1 3+ ~ 3) - 15 2 -6 ( b ) 12x - 310 Graph each inequalitjr and shade the solution region. X l])! . . . . . The method of proof holds in the general case. l ( b ) and (e). . . 17-22 Fig. . . SOLUTION (a) -(9-x2), 4 y = -+ ; 9 Note that y is real when 9 - x2 L 0, i.e., when -3 ( x or less than -3 are excluded. 24.26 A bank tried to attract new, large, long-term investments by paying 9.75% interest compounded continuously if at least $30 OOO was invested for at least 5 years. . . B. Permutations with some things alike, taken all at a time The number of permutations P of n things taken all at a time, of which n1 are alike, n2 others are alike, n3 others are alike, etc., is P= n! . ( a ) center at the origin and goes through (2,6) (6) ends of diameter at (-7,2) and (5,4) SOLUTION + ( a ) The standard form of a circle with center at the origin is x2 y2 = 3. Thus the cofactor corresponding to the given element is 29.16 Write the minors and cofactors of the elements in the 4th row of the determinant 3 - 2 4 2 2 1 5 - 3 1 5 - 2 2 -3 -2 -4 1 SOLUTION The elements in the 4th row are -3, - t, -4, 1. The minimum for C(x,y), if it exists, will occur at point A, B, C, or D,so we evaluate the obective function at these points. . ( x + 3)2 (y - 4)2 +-=l 7 9 7 - 1, center (5, l ) , vertices (7,l) and (3, l ), foci (6, l ) and (4, l ), covertices (5,1+ fi)and (5,l - fi) - 1, center (2, -2), vertices (2,2) and (2, -6), foci ( 2 , l ) and (2, -9, covertices (2 + ~, -2) and (2 -fi, -2) (y + 5 ) 2 (x + 3)2 +-- 8 - 1, center (-3, - 5 ) , vertices (-3, -2) and (-3, -8), foci (-3, -4) and (-3, -6), 9 covertices (-3 + 2t/jl, -5) and (-3 - 2 ~-5) , 5 (y+3)2 -_-4 (') 3 ~ 16 + 4 (a) -x2_ - - y2 -1 9 16 y2 x2 (6) - - - = 1 64 36 (U) - 3)2 = 2001 - 5) + ( ~ - 5 ) 0 ~ - 1)2 4 (X 0,+ a2 (x - 3)2 (d) 1 6 4 y2 x2 (6) -+-=1 25 9 17.18 (6) = 1, center (3, l), vertices (3 + fi,1) and (3 - fi,l ), foci (4, l ) and (2, l ) , covertices (3,3) and (3, -1) 0, - 2)2 ---- (x (y - 1)2 (d) ---= (x (c) 1 9 - 4)2 8 + 1)2 25 -1 1 (X- 1)2 - 1, center (1, -3), vertices (1, -1) and (1, - 5 ) , foci (1,O) and (1, -8) 5 f12-M2= 1, center (1,3), vertices (-1,3) 5 4 and (3,3), foci (4,3) and (-2,3) ( c ) -_-(x - 3)2 4 (y + 5)2 - 1, center (3, - 5 ) , vertices (5, -5) and (1, - 5 ) , foci (7, -5) and (-1, -5) 12 (d) -_-0,-3)2 ( x+ 1)2 - 1, center (-1,3), vertices (-1,7) and (-1, -l), foci (1,3 +2%'3) and 4 (1,3-2t/5) 16 Chapter 18 Systems of Equations Involving Quadratics 18.1 GRAPHICAL SOLUTION The real simultaneous solutions of two quadratic equations in x and y are the values of x and y corresponding to the points of intersection of the graphs of the two equations. . . Hence the required number of ways = 4C1 + 4C2+ 4C3+ 4C4= 4 + 6 + 4 + 1 = 15 ways. . . . Note that the proof is essentially a reversal of the steps in the first paragraph. . . . Schaum’s is the key to faster learning and higher grades in every subject. . . Interchanging two adjacent rows results in the interchange of two adjacent subscripts in each term of the expansion. . . SOLUTION Since the numbers are between 3000 and 5000, they consist of 4 digits. EXAMPLE 24.6. . . . Terms . . . Studyworks calls this a range of values. . - 2 4 = 2.4 = 8 3 5 3.5 15' -*- 3 8 =3 - 8 - 24 -.4 9 2 4-9-36=3 (6) The reciprocal of a fraction is a fraction whose numerator is the denominator of the given fraction and whose denominator is the numerator of the given fraction. . . 1 + 26.2 EXPANSION OF (a x)" If n is a positive integer we expand (a + x ) as ~ shown below: This equation is called the binomial theorem, or binomial formula. (x - h)2 a2 b2 or The vertices are (h + a, k) and (h - a , k), so ( h + a , k ) = (1 + a , -1) = ( 5 , -1). . . . . 4+(-2)]=[: -1+2 + :] The matrix -A is called the opposite of matrix A and each entry in -A is the opposite of the corresponding entry in A. . a . 8 ~ = 2 . . The equation of the circle is ( x + 1)2 + 0, - 3)2 = 37. . Then the problem is to find the number of ways A can select 3 maps out of 12, not including the selection by A of his original three maps. Helpful tables and illustrations increase your understanding of the subject at hand. 4x2- 28 x4+x2-6 Hence - 8 x2+3 4 x2-2- Supplementary Problems Find the partial fraction decomposition of each rational fraction. . 19.27 Prove that x y + l z x + y if x l l a n d y r l or if x s l a n d y l l . In (1) the transverse axis V V ‘ lies on the x axis, the vertices are V ( a ,0) and V ‘ ( - a , O ) , and the foci are at F(c,O) and F’(-c,O) (see Fig. 6 - 5 - 4= 6720 words. The solution of the system of equations can be aided by the use of a graphing calculator, especially when using the matrix methods discussed in Chapter 30. He is one of Schaum’s most prolific authors. . . [AII]= [ 1 3 4 1 0 0 ] [ , 3 4 l o o ] -2 -5 -3 0 1 0 -R2+2R1 0 1 5 2 1 0 1 4 9 0 0 1 R 3 - R 1 0 1 5 - 1 0 1 R1-3R2 5 R3-RZ 1 0 -11 -5 -3 0 [ o l 5 2 1 0 ] 0 0 0 -3 -1 1 The matrix A is row equivalent to the matrix on the left. . . . . . . = n.(n - I)! Thefociareat ( h + c , k ) a n d ( h - ~ , k ) , s o ( h + c , k ) = ( l , 2 ) a n d- 5 + c = 1, with 184 CONIC SECTIONS [CHAP. . Professor of Mathematics Fort Valley State University SCHAUM’S OUTLINE SERIES McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogotci CaracuJ Lisbon London Madrid Mexico City Milan Montreal New Dehli San Juan Singapore Sydney Tokyo Toronto MURRAY R. SPIEGEL received the M.S.degree in Physics and the Ph.D. in Mathematics from Cornell University. . (1) Linear factors none of which are repeated If a linear factor ax + b occurs once as a factor of the denominator of the given fraction, then corresponding to this factor associate a partial fraction A/(ax+ b), where A is a constant # 0. (a) 42 + 23, 23 + 42 (f) 35.28 (i) 7 2 + 2 4 + 6 4 + 1 6 (b) 27 + (48 + 12), (27 + 48) + 12 (8) 756 + 21 ( j ) 4 + 2 +6 s 3 -2 + 2 +3.4 (c) 125 - (38 + 27) (40 + 21)(72 - 38) ( k ) 128 + (2.4), (128 + 2) ‘4 (d) 6 . . = 91 ways. . . . . . . . . . . . 25.63 How many even numbers of four different digits each can be formed from the digits 3,5,6,7,9? Unless otherwise specified we shall deal with real numbers. . . 4= The number of ways in which 4 persons can take their places in a cab having 6 seats is 6P4 = 6 . 25.93 How many words of 2 vowels and 3 consonants may be formed (considering any set a word) from the letters of the word ( a ) stenographic, (6) facetious? SOLUTION Interest I = Prt = 400(0.03)(2) = $24. . = n(n - l)! . More than 40 million students have trusted Schaum’s to help them succeed in the classroom and on exams. . . . 191 ( b ) If vz>-a2ab +b’ then ab >- 4a2b2 ( a + b)2 * ( a + b ) 2> 4ab and (a - b ) 2> 0 which is true if a # b . . . . 7 . Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any forms or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. . The midpoint M of the line segment having endpoints (xl,yl) and ( x 2 , y 2 ) is Thus, the center is The radius of a circle is the distance from the center to the endpoint of the diameter. . . . . . Write the equation of the ellipse 18x2+ 12y2- 144x + 48y + 120 = 0 in standard form. . . . SOLUTION ( a ) The first prize can be awarded in 10 different ways and, when it is awarded, the second prize can be given in 9 ways, since both prizes may not be given to the same contestant. + 2u3 6x2 - 9 is an improper fraction. X-6 Check Using Study Works: (Remember the real check must be t o use the original equation.) . . . . SOLUTION Let D be the value of the determinant. 3!2! . . SOLUTION One or more ties may be selected in (28 - 1) ways. . . . \Algebra and Geometry", D. Holten and J. Lloyd, CBRC, 1978. . Then the men can be arranged in 3! . . . . Only 1 left in stock - order soon. Since the foci are on the x axis, the standard form is x2 -+- y2 a2 b2 The equation of the ellipse is x2 y2 -+3 2 =1. . . . . . The standard form equations for central hyperbolas are: y2 (1) x2 ---= a2 b2 y2 x2 (2) a2 - -62- and 1 The distance from the center to a vertex is denoted by a and the distance from the center to a focus is c. For a hyperbola, c’ = a2 + b2 and b is a positive number. Schaum’s Outline of Theory and Problems of COLLEGE ALGEBRA Copyright @ 1998, 1956 by The McGraw-Hill Companies, Inc. All rights reserved. SOLUTION There are 3 . . 17-16 (b) hyperbola, Fig. . . . . . . 3! .+#+' + x ) k + l = a k + l + ( k + 1)akx + - + ( k + l ) k ( k(r- l-) .l)! SOLUTION Each appliance may be dealt with in 2 ways, as it can be chosen or not chosen. abc = a(6c) = (ab)c, 3.4.6 = 3(4.6) = (3.4)6 = 72 Distributive property for multiplication over addition The product of a number a by the sum of two numbers (6 + c) is equal to the sum of the products ab and ac. . From a2 = b2 + c2, we get c = 6. . As you can see on the following pages, all the math appears in familiar notation, including units. . . 2 nP4 = 30enP4. . A = P + I = $1744.90. . The symbols for “greater than” and “less than” are > and < respectively. . . . . Note: 3lr is a proximately 3(3.14) = 9.42, so that the corresponding point is between +9 and +10 as indicated. . . Not Enough Time? . 29.27 Find non-trivial solutions for the system + 3y - 22 = 0 2x - 4y + z = 0 x+y-z=o x if they exist. . + A - B X C C1j C2j ... ... Cij Cmj ... Cmn EXAMPLE 30.3. Then the remaining 6 digits may be arranged in the 3 other places in 6P3 ways. . . . . An infinite number of solutions is obtained by assigning various values to z. . = n(n - l ) ( n - 2) - n(n - l ) ( n- 2) - - 2 . . (U) (6) y 2 + 3~ - 6y = 0 x2 - 4~ - 12y - 32 = 0 ( a ) x2 - 4~ - 12y - 32 = 0 x2 - 4~ = 12y + 32 x2 - 4x + 4 = 12y + 32 + 4 ( X - 4)2 = 12y 36 ( x - 4)2 = 12(y + 3 ) ( b ) y2 + 3x - 6y = 0 y2 - 6y = - 3 ~ y2- 6y + 9 = - 3 + ~9 (y - 3)2 = -3(x - 3 ) + 17.5 reorganize terms complete the square for x factor right-hand side of equation standard form reorganize terms complete the square for y standard form ELLIPSES An ellipse is the locus of all points in a plane such that the sum of the distances from two fixed points, the foci, to any point on the locus is a constant. The transverse axis has length 4 so 2a = 4 and a = 2. How many times the hearing threshold intensity is the intensity of a relatively quiet room? Thus, for A=[-'-2 2 0 I:- -A = -20 -131 Multiplying a matrix by a scalar (real number) results in every entry in the matrix being multiplied by the scalar. . Find its dimensions. The product of the factors (x - 4) and (x + 4) is negative. 17-14 4 y 2 = - ( x 2 - 9), 9 Note that x cannot have a valrle between -3 and 3 if y is to be rea I. eBook Shop: Schaum's Outline Series: Schaum's Outline of College Algebra, Third Edition von Robert E. Moyer als Download. . + k2 + ( k + = k(k + 1)(2k ' ) + ( k + 6 + 368 MATHEMATICAL INDUCTION [CHAP 31 k(k + 1)(2k + 1) + 6(k + 1)* 6 - (k + 1)[(2k2 + k) + (6k + 6)L - l k + l ) ( k + 2)(2k + 3) 6 6 The right hand side of this equation = + which is the value of n(n + 1)(2n + 1)/6 when n is replaced by (k 1). . 8:51 AM . . Elementary Row Operations Interchange two rows. . . Algebra Moderna Schaum - Frank.Ayres. . . . . . NINTH EDITION . . . . n! . CHAP 291 DETERMINANTS OF ORDER n 351 29.36 For the determinant 2 3 -2 -1 -2 2 -3 1 2 -1 2 4 - 1 3 -1 4 (a) write the minors and cofactors of the elements in the 3rd row, (6) express the value of the determinant in terms of minors or cofactors, ( c ) find the value of the determinant. . . . . Absolute value is indicated by two vertical lines surrounding the number. The foci have coordinates F(h + c, k) and F'(h - c, k), the vertices are at V(h + a , k ) and V ' ( h - a, k), and the 172 CONIC SECTIONS [CHAP: 17 I B'(0, -b) Fig. If [H+] is the concentration of hydrogen ions in moles per liter, the pH is given by the formula: pH = -log[H+] EXAMPLE 24.8. . . . . . Thus, 1 + a = 5 and a = 4. SOLUTION A = AOevrf 0.01Ao = Aoe-0.000124r 0.01 = e-0.000124f lnO.O1 = -0.OOO 124t Ine In (1 X 10-2) = -0.O00 12441) 0 - 2(2.3026) = -0.OOO 124t -4.6052 = -0.0oO 124t 37 139 = t t = 37 100 years 24.20 The population of the world increased from 2.5 billion in 1950 to 5.0 billion in 1987. . Note. . Check the solutions obtained. SAMPLE SCREENS APPENDIX C] 307 6 ) The reciprocal of a fraction is a fraction whose numerator is the denominator of the given fraction and whose denominator is the numerator of the given fraction. . . . . The transverse axis joins the vertices, so its length is 2a, so 2u = 4 and a = 2. . . . - 19.12 Prove: if if (a) x 2 - y 2 > x - y (6) x 2 - y 2 < x - y x+y>l x+y>l and and x>y x y , x - y > 0. (6) I = Prt = 1562.60(0.035)(10/3) = $182.30. . The following formula is very useful in simplifying calculations: ncr =ncn-r. . To find C, let x = 2: 4 - 1 2 + 2 = 4 C , C = -3/2. . . . Home. . . 25.13 In how many ways can n women be seated in a row so that 2 particular women will not be next to each other? . SOLUTION Total number of ways in which 3 can be selected out of 25 = 25c3. . . Schaum’s Outline of College Algebra pdf Schaum’s Outline of College Algebra pdf : Pages 464 By Murray R. Spiegel and Robert E. Moyer Publisher: McGraw-Hill Education, Year: 2018 ISBN: 1260120767, 9781260120769 Search in Amazon.com Description: Tough Test Questions? . . . . . Absolute Value Inequalities . . 29.9 Prove Property VII: If to each element of a row (or column) of a determinant is added m times the corresponding element of any other row (or column), the value of the determinant is not changed. Thus the number of inversions of subscripts is either increased by one or decreased by one. 6.3 Complex Fractions . . . Fundamental Operations with Algebraic … . . . . . . . The asymptote is a line that the graph of the hyperbola approaches but does not reach. . . . . . (a) Evaluate each factorial. . 24.40 A relatively quiet room has a background noise level of 32 decibels. . . 17-11). . . . . . . . Thus, 69!can be displayed and 70! . . )- k)2 -( X - hI2 7 - 3 b2 + In (3) the major axis is parallel to the x axis and the minor axis is parallel to the y axis. . . . (7) To divide two fractions, multiply the first by the reciprocal of the second. . Thus 3 + 4 = 7 , (-3)+(-4)=-7. Otherwise. . . . . Sorry, preview is currently unavailable. . 2 . . . 5 ' 4 . . . . Step 2. . . . For example, a woman has in her pocket a quarter, a dime, a nickel, and a penny. . (6) What would be the number of inversions in the letters of bld3~2a4when the subscripts are arranged in natural order? . . . 2*1.10! - -10-9.8 -- 120. . . . . . . . . . . . . . . . . . To add two numbers with unlike signs, find the difference between their absolute values and prefix the sign of the number with greater absolute value. THE BINOMIAL THEOREM . . . Number of selections = (Z5- l)(z4 - 1)(23) = 31 - 15.8 = 3720 selections. 6 Step 2 . . . . . . . . . . . \Schaum’s Outline of Linear Algebra", S. Lipschutz and M. Lipson, McGraw-Hill 2008. . A. . . . 263 23.1 23.2 23.3 23.4 23.5 23.6 23.7 Chapter 24 APPLICATIONS OF LOGARITHMS AND EXPONENTS . 2 . . . + 19.15 Determine graphically the range of values of x defined by (a) x 2 + 2 x - 3 = 0 (b) X2+2x-3>0 (c) x2 + 2x - 3 CO. These books contain lots of information and tutorials to improve your knowledge, available for all levels! 25.26 In how many ways can 4 men and 4 women be seated at a round table if each woman is to be between two men? . and (2) Whenever for n = k P ( k ) is true implies P(k + 1) is true. . . SOLUTlON To eliminate y, multiply (1) by 2, (2) by 3 and add; then 13x2 = 117, x2 = 9, x = +_3. The covertices are ( 0 , b ) and (0, - b ) , so B(0,8/9) and B'(0, -8/9). Now put x = 2 or x = -2 in (1) and obtain y = +1. . . 24 rounded to thousandths. . 10! ): zero, 49 Extraneous roots, solutions, 74 Extremes, 81 Factor, 31 greatest common, 33 prime, 31 Factor theorem, 210 Factorial notation, 287 Factoring, 31 Failure, probability of, 311 Formulas, 74 Fourth proportional, 81 Fractional exponents, 49 Fractions, 17-18, 4 1 4 3 complex, 43 equivalent, 41 improper, 372 operations with, 4 partial, 372 proper, 372 rational algebraic, 41 reduced to lowest terms, 41 signs of, 4 Function, 89 graph of, 90-94 linear, 75, 127-130 notation for, 90 polynomial, 210 quadratic, 75, 149-152 Fundamental Counting Principle, 287 Fundamental Theorem of Algebra, 241 General or nth term, 241 Geometric mean, 242 Geometric means, 242 Geometric sequence, 241 infinite, 242 Geometric series, infinite, 242 Graphical solution of equations, 136, 188 Graphs, 89-95 of equations, 89-95, 136, 167-178 of functions, 90-94 of linear equations in two variables, 136 of quadratic equations in two variables, 188 with holes, 231 Greater than, 2 Greatest common factor, 33 Grouping, symbols of, 13 Grouping of terms, factoring by, 32 Harmonic mean, 242 Harmonic means, 242 Harmonic sequence, 242 Holes, in graph, 231 Homogeneous linear equations, 338 Hyperbola, 175 i, 67 Identically equal polynomials, 372 INDEX Identity, 73 matrix, 355 property, 22 Imaginary numbers, 2, 67 Imaginary part of a complex number, 67 Imaginary roots, 151 Imaginary unit, 2, 67 Improper fraction, 372 Inconsistent equations, 136 Independent events, 311 Independent variable, 89 Index, 48, 58 Index of a radical, 58 reduction of, 59 Induction, mathematical, 366 Inequalities, 195 absolute, 195 conditional, 195 graphical solution of, 197, 198 higher order, 196 principles of, 195 sense of, 195 signs of, 195 Infinite geometric sequence or series, 242 Infinity, 242 Integers, 22 Integral roots theorem, 212 Interest, 274 compound, 274 simple, 274 Intermediate Value Theorem, 212 Interpolation in logarithms, 260 Interpolation, linear, 260 Inverse property, 22 Inversions, 335 Irrational number, 1 Irrational roots, 151 approximating, 213 Irrationality, proofs of, 78, 221 Inverse matrix, 356 Inverse property, 22 Least common denominator, 42 Least common multiple, 33 Less than, 2 Like terms, 12 Linear equations, 113 consistent, 136 dependent, 136 determinant solution of system of, 325-338 graphical solution of systems of, 136 homogeneous, 338 inconsistent, 136 in one variable, 113 simultaneous, systems of, 136 Linear function, 113 Linear interpolation, 260 Linear programming, 199 Lines, 127 intercept form, 130 slope-intercept form, 128 slope-point form, 129 403 Lines (Cont. . . . . . . Applications Biology and Life Sciences Agriculture, 495, 767, 777, 995, 1020 Air sacs in the lungs, 28 Alligator length, SCHAUM’S OUTLINE OF THEORY AND PROBLEMS of COLLEGE ALGEBRA Second Edition MURRAY R. SPIEGEL, Ph.D. Former Profdsor and Chairman Mathematics Department Rensselaer Polytechnic Institute Hartford Graduate Center ROBERT E. MOYER, Ph.D. . Hence ( a - 6) + ( c - d ) > 0, (a c ) - (6 + d ) > 0 and ( a + c) > (6 + d ) . . . Since the last row results in the equation Oz = 1, which has no solution, the system of equations has no solution. . . . . . . . Zero is considered a rational number without sign. Find the lengths of the two legs. Substituting in the quadratic equation, y2 - y( 4 - 2y) = 7, 3g - 4y - 7 = 0, 0, + 1)(3y - 7) = 0 and y = - 1, 7/3. . Since each of the 2 ways of dealing with an appliance is associated with 2 ways of dealing with each of the other appliances, the number of ways of dealing with the 4 appliances = 2.2.2.2 = 24 ways. . Write the equation of the hyperbola that has the given characteristics. Then, since it holds for n = 2, it holds for n = 2 + 1 = 3, and so on. . . 171 Hyperbolas . . . . . . Hence the number of ways in which 9 books can be arranged on a shelf so that 3 specified books are never all 3 together = 362 880 - 30 240 = 332 640 ways. . SOLUTION A = P(l+rt) 24.5 A-P r=-Pt or - 1OOO-800 800(5 ) = 0.05 or 5%. . . Since you can not have a company process a negative number of tons of paper, x L 0 and y 2 0 . . 24.47 Radioactive strontium-90 is used in nuclear reactors and decays exponentially with an annual rate of decay of 2.48%. . . . . . . . . 17-20 5-- -5 t Fig. Hence the required number of arrangements = sP5 4P4= S!4!= 120 - 24 = 2880. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 2 1 0 9 ISBN 0-07-060266-2 Sponsoring Editor: Barbara Gilson Production Supervisor: Tina Cameron Editing Supervisor: Maureen B . . . Method 2 . . ways. . . . . . 25.1 25.2 25.3 25.4 Chapter 26 Definition of a Logarithm . . . . . . Schaums Outlines Elementary Algebra. . Pages 416 . . 362 MATRICES [CHAP 30 SOLUTION [:: I[: : ][;I :][;I=[: The solution to the system is (4, 8). 136 15.2 Systems of Three Linear Equations . . . . . . [ :] -2 30.6 If A = -1 - (a) 2 X + 3 A = B B= and -4 . 181 CONIC SECTIONS CHAP. . . . . . . . . x+2 ---1113 2x2-7~-15-2x+3 -1 +--X7/13 + 7 - 5 - 13(2~+3) 13(~-5)' Another method. . . . . 170! A = Aoe-'' where A. is the initial amount, t is the annual rate of decay or decline, t is the time in years, and A is the ending amount. where n and r are integers and r In. A partial list is given . The first equation is y = f(x) + 1. . ;permutations, nPr, and combinations, nCr.As factorials get larger, the results are displayed in scientific notation. CHAP. . foci are ( - 1 , O ) and (-1,O) and length of minor axis is 2 (6) vertices are at (5, -1) and (-3, -1) and c = 3. . . 29.33 Show that the determinant 1 2 3 2 4 6 3 8 1 2 4 16 24 4 3 2 1 equals zero. . 17.8 Write the equation of the ellipse in standard form and determine its center, vertices, foci, and covert ices.

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