 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 20077 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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def 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.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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