 3.6.1: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.2: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.3: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.4: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.5: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.6: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.7: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.8: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.9: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.10: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.11: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.12: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.13: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.14: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.15: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.16: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.17: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.18: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.19: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.20: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.21: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.22: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.23: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.24: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.25: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.26: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.27: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.28: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.29: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.30: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.31: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.32: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.33: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.34: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.35: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.36: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.37: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.38: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.39: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.40: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.41: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.42: Solve each polynomial inequality in Exercises 142 and graph the so...
 3.6.43: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.44: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.45: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.46: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.47: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.48: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.49: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.50: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.51: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.52: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.53: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.54: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.55: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.56: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.57: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.58: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.59: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.60: Solve each rationat inequality in Exercises 4360 and graph the sol...
 3.6.61: In Exercises 6164, find the domain of eadz fimction. f(x)  V2t 1 ...
 3.6.62: In Exercises 6164, find the domain of eadz fimction. f(x)  :r.=::...
 3.6.63: In Exercises 6164, find the domain of eadz fimction. fl.x)  /.2!....
 3.6.64: In Exercises 6164, find the domain of eadz fimction. f(x)  /_ x_ ...
 3.6.65: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.66: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.67: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.68: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.69: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.70: Solve each inequality in Exercises 6570 and graph the solution set...
 3.6.71: In Exercises 7172. use the graph of the polynomial ftmction 10 sol...
 3.6.72: In Exercises 7172. use the graph of the polynomial ftmction 10 sol...
 3.6.73: In Exercises 7374, use the graph of the rational function to solve...
 3.6.74: In Exercises 7374, use the graph of the rational function to solve...
 3.6.75: Use the position function >(1)   16r' + . , + ... ( <w  inotial ...
 3.6.76: Use the position function >(1)   16r' + . , + ... ( <w  inotial ...
 3.6.77: The functions and f{.:c)  0.0875..'  0.4x + 66.6 . ..,,..~ g(.:c...
 3.6.78: The functions and f{.:c)  0.0875..'  0.4x + 66.6 . ..,,..~ g(.:c...
 3.6.79: The perimeter of a rectangle is 50 feet. Describe the possible leng...
 3.6.80: The perimeter of a rectangle is 180 feet. Dcscribe the possible len...
 3.6.81: What is a polynomial inequality?
 3.6.82: What is a rational inequality?
 3.6.83: If f is a polynomial or rational runction. explain how the graph of...
 3.6.84: Use a graphing utility to verify your solution sets to any three of...
 3.6.85: Use a graphing utility to verify) our solution sets to any three of...
 3.6.86: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.87: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.88: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.89: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.90: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.91: Solve each inequality in Exercises 8691 using a graphing utility. ...
 3.6.92: The graph shows stopph1g distances for truck.f nt ''arious speed.s ...
 3.6.93: The graph shows stopph1g distances for truck.f nt ''arious speed.s ...
 3.6.94: In Exercises 9497, determine whether each statement makes sense or...
 3.6.95: In Exercises 9497, determine whether each statement makes sense or...
 3.6.96: In Exercises 9497, determine whether each statement makes sense or...
 3.6.97: In Exercises 9497, determine whether each statement makes sense or...
 3.6.98: In Exercises 98101. determine whether each statement is true or fo...
 3.6.99: In Exercises 98101. determine whether each statement is true or fo...
 3.6.100: In Exercises 98101. determine whether each statement is true or fo...
 3.6.101: In Exercises 98101. determine whether each statement is true or fo...
 3.6.102: Write a polynomiaJ inequality whose solution set is ( 3, 5J.
 3.6.103: Write a rational inequality whose solution set is ,  4) u (3, ).
 3.6.104: In Exercises 104107. use hJSpection to describe each inequality's ...
 3.6.105: In Exercises 104107. use hJSpection to describe each inequality's ...
 3.6.106: In Exercises 104107. use hJSpection to describe each inequality's ...
 3.6.107: In Exercises 104107. use hJSpection to describe each inequality's ...
 3.6.108: The graphing utility screen shows the graph of ."  4t'  8x + 7. ...
 3.6.109: The graphing utility screen shows the graph of y  V27 3x2 Write an...
 3.6.110: Exercises 110112 help you prepare for tire material c.overed in t...
 3.6.111: Exercises 110112 help you prepare for tire material c.overed in t...
 3.6.112: Exercises 110112 help you prepare for tire material c.overed in t...
Solutions for Chapter 3.6: Polynomial and Rational Inequalities
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 3.6: Polynomial and Rational Inequalities
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 6. College Algebra was written by and is associated to the ISBN: 9780321782281. Since 112 problems in chapter 3.6: Polynomial and Rational Inequalities have been answered, more than 39121 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.6: Polynomial and Rational Inequalities includes 112 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).