 8.5.1: We solve the polynomial inequality x2 + 8x + 15 7 0 by first solvin...
 8.5.2: The points at 5 and 3 shown above divide the number line into thr...
 8.5.3: True or false: A test value for the leftmost interval on the number...
 8.5.4: True or false: A test value for the rightmost interval on the numbe...
 8.5.5: Consider the rational inequality x  1 x + 2 0. Setting the numerat...
 8.5.6: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.7: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.8: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.9: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.10: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.11: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.12: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.13: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.14: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.15: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.16: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.17: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.18: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.19: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.20: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.21: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.22: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.23: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.24: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.25: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.26: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.27: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.28: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.29: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.30: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.31: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.32: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.33: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.34: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.35: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.36: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.37: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.38: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.39: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.40: Solve each polynomial inequality in Exercises 140 and graph the sol...
 8.5.41: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.42: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.43: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.44: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.45: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.46: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.47: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.48: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.49: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.50: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.51: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.52: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.53: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.54: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.55: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.56: Solve each rational inequality in Exercises 4156 and graph the solu...
 8.5.57: In Exercises 5760, use the given functions to find all values of x ...
 8.5.58: In Exercises 5760, use the given functions to find all values of x ...
 8.5.59: In Exercises 5760, use the given functions to find all values of x ...
 8.5.60: In Exercises 5760, use the given functions to find all values of x ...
 8.5.61: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.62: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.63: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.64: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.65: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.66: Solve each inequality in Exercises 6166 and graph the solution set ...
 8.5.67: In Exercises 6768, use the graph of the polynomial function to solv...
 8.5.68: In Exercises 6768, use the graph of the polynomial function to solv...
 8.5.69: In Exercises 6970, use the graph of the rational function to solve ...
 8.5.70: In Exercises 6970, use the graph of the rational function to solve ...
 8.5.71: You throw a ball straight up from a rooftop 160 feet high with an i...
 8.5.72: Divers in Acapulco, Mexico, dive headfirst from the top of a cliff ...
 8.5.73: a. Use the given functions to find the stopping distance on dry pav...
 8.5.74: a. Use the given functions to find the stopping distance on dry pav...
 8.5.75: Describe the companys production level so that the average cost of ...
 8.5.76: Describe the companys production level so that the average cost of ...
 8.5.77: The perimeter of a rectangle is 50 feet. Describe the possible leng...
 8.5.78: The perimeter of a rectangle is 180 feet. Describe the possible len...
 8.5.79: What is a polynomial inequality?
 8.5.80: What is a rational inequality?
 8.5.81: Describe similarities and differences between the solutions of (x ...
 8.5.82: Solve each inequality in Exercises 8287 using a graphing utilityx2 ...
 8.5.83: Solve each inequality in Exercises 8287 using a graphing utility2x2...
 8.5.84: Solve each inequality in Exercises 8287 using a graphing utility x ...
 8.5.85: Solve each inequality in Exercises 8287 using a graphing utility x ...
 8.5.86: Solve each inequality in Exercises 8287 using a graphing utility 1x...
 8.5.87: Solve each inequality in Exercises 8287 using a graphing utility x3...
 8.5.88: The graph shows stopping distances for trucks at various speeds on ...
 8.5.89: The graph shows stopping distances for trucks at various speeds on ...
 8.5.90: Make Sense? In Exercises 9093, determine whether each statement mak...
 8.5.91: Make Sense? In Exercises 9093, determine whether each statement mak...
 8.5.92: Make Sense? In Exercises 9093, determine whether each statement mak...
 8.5.93: Make Sense? In Exercises 9093, determine whether each statement mak...
 8.5.94: In Exercises 9497, determine whether each statement is true or fals...
 8.5.95: In Exercises 9497, determine whether each statement is true or fals...
 8.5.96: In Exercises 9497, determine whether each statement is true or fals...
 8.5.97: In Exercises 9497, determine whether each statement is true or fals...
 8.5.98: Write a quadratic inequality whose solution set is [3, 5]
 8.5.99: Write a rational inequality whose solution set is ( , 4) [3, ).
 8.5.100: In Exercises 100103, use inspection to describe each inequalitys so...
 8.5.101: In Exercises 100103, use inspection to describe each inequalitys so...
 8.5.102: In Exercises 100103, use inspection to describe each inequalitys so...
 8.5.103: In Exercises 100103, use inspection to describe each inequalitys so...
 8.5.104: The graphing calculator screen shows the graph of y = 4x2  8x + 7....
 8.5.105: The graphing calculator screen shows the graph of y = 227  3x2 . W...
 8.5.106: Solve: 2 x  5 3 2 6 8. (Section 4.3, Example 4)
 8.5.107: Divide: 2x + 6 x2 + 8x + 16 , x2  9 x2 + 3x  4 . (Section 6.1, Ex...
 8.5.108: Factor completely: x4  16y4 . (Section 5.5, Example 3)
 8.5.109: Exercises 109111 will help you prepare for the material covered in ...
 8.5.110: Exercises 109111 will help you prepare for the material covered in ...
 8.5.111: Exercises 109111 will help you prepare for the material covered in ...
Solutions for Chapter 8.5: Polynomial and Rational Inequalities
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 8.5: Polynomial and Rational Inequalities
Get Full SolutionsChapter 8.5: Polynomial and Rational Inequalities includes 111 full stepbystep solutions. Since 111 problems in chapter 8.5: Polynomial and Rational Inequalities have been answered, more than 53326 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.