 6.6.1: We clear a rational equation of fractions by multiplying both sides...
 6.6.2: We reject any proposed solution of a rational equation that causes ...
 6.6.3: The first step in solving 4 x + 1 2 = 5 x is to multiply both sides...
 6.6.4: The first step in solving x  6 x + 5 = x  3 x + 1 is to multiply ...
 6.6.5: The restrictions on the variable in the rational equation 1 x  2 ...
 6.6.6: 5 x + 4 + 3 x + 3 = 12x + 9 (x + 4)(x + 3) (x + 4)(x + 3)a 5 x + 4 ...
 6.6.7: True or false: A rational equation can have no solution. __________...
 6.6.8: 6 x + 3 = 4 x  3
 6.6.9: x  6 x + 5 = x  3 x + 1
 6.6.10: x + 2 x + 10 = x  3 x + 4
 6.6.11: x + 6 x + 3 = 3 x + 3 + 2
 6.6.12: 3x + 1 x  4 = 6x + 5 2x  7
 6.6.13: 1  4 x + 7 = 5 x + 7
 6.6.14: 5  2 x  5 = 3 x  5
 6.6.15: 4x x + 2 + 2 x  1 = 4
 6.6.16: 3x x + 1 + 4 x  2 = 3
 6.6.17: 8 x2  9 + 4 x + 3 = 2 x  3
 6.6.18: 32 x2  25 = 4 x + 5 + 2 x  5
 6.6.19: x + 7 x = 8
 6.6.20: x + 6 x = 7
 6.6.21: 6 x  x 3 = 1
 6.6.22: x 2  12 x = 1
 6.6.23: x + 6 3x  12 = 5 x  4 + 2 3
 6.6.24: 1 5x + 5 = 3 x + 1  7 5
 6.6.25: 1 x  1 + 1 x + 1 = 2 x2  1
 6.6.26: 1 x  2 + 1 x + 2 = 4 x2  4
 6.6.27: 5 x + 4 + 3 x + 3 = 12x + 19 x2 + 7x + 12
 6.6.28: 2x  1 x2 + 2x  8 + 2 x + 4 = 1 x  2
 6.6.29: 4x x + 3  12 x  3 = 4x2 + 36 x2  9
 6.6.30: 2 x + 3  5 x + 1 = 3x + 5 x2 + 4x + 3
 6.6.31: 4 x2 + 3x  10 + 1 x2 + 9x + 20 = 2 x2 + 2x  8
 6.6.32: 4 x2 + 3x  10  1 x2 + x  6 = 3 x2  x  12
 6.6.33: 3y y2 + 5y + 6 + 2 y2 + y  2 = 5y y2 + 2y  3
 6.6.34: y  1 y2  4 + y y2  y  2 = 2y  1 y2 + 3y + 2
 6.6.35: g(x) = x 2 + 20 x ; g(a) = 7
 6.6.36: g(x) = x 4 + 5 x ; g(a) = 3
 6.6.37: g(x) = 5 x + 2 + 25 x2 + 4x + 4 ; g(a) = 20
 6.6.38: g(x) = 3x  2 x + 1 + x + 2 x  1 ; g(a) = 4
 6.6.39: x + 2 x2  x  6 x2  1
 6.6.40: x + 3 x2  x  8 x2  1
 6.6.41: x + 2 x2  x  6 x2  1 = 0
 6.6.42: x + 3 x2  x  8 x2  1 = 0
 6.6.43: 1 x3  8 + 3 (x  2)(x2 + 2x + 4) = 2 x2 + 2x + 4
 6.6.44: 2 x3  1 + 4 (x  1)(x2 + x + 1) =  1 x2 + x + 1
 6.6.45: 1 x3  8 + 3 (x  2)(x2 + 2x + 4)  2 x2 + 2x + 4
 6.6.46: 2 x3  1 + 4 (x  1)(x2 + x + 1) + 1 x2 + x + 1
 6.6.47: f(x) = x + 2 x + 3 , g(x) = x + 1 x2 + 2x  3
 6.6.48: f(x) = 4 x  3 , g(x) = 10 x2 + x  12
 6.6.49: f(x) = 5 x  4 , g(x) = 3 x  3 , h(x) = x2  20 x2  7x + 12
 6.6.50: f(x) = 6 x + 3 , g(x) = 2x x  3 , h(x) =  28 x2  9
 6.6.51: If the government commits $375 million for this project, what perce...
 6.6.52: If the government commits $750 million for this project, what perce...
 6.6.53: After how many days do the students remember 8 words? Identify your...
 6.6.54: After how many days do the students remember 7 words? Identify your...
 6.6.55: What is the horizontal asymptote of the graph? Describe what this m...
 6.6.56: According to the graph, between which two days do students forget t...
 6.6.57: How many learning trials are necessary for 0.95 of the responses to...
 6.6.58: How many learning trials are necessary for 0.5 of the responses to ...
 6.6.59: Describe the trend shown by the graph in terms of learning new task...
 6.6.60: What is the horizontal asymptote of the graph? Once the performance...
 6.6.61: A company wants to increase the 10% peroxide content of its product...
 6.6.62: Suppose that x liters of pure acid are added to 200 liters of a 35%...
 6.6.63: What is a rational equation?
 6.6.64: Explain how to solve a rational equation.
 6.6.65: Explain how to find restrictions on the variable in a rational equa...
 6.6.66: Why should restrictions on the variable in a rational equation be l...
 6.6.67: Describe similarities and differences between the procedures needed...
 6.6.68: Rational functions model learning and forgetting. Use the graphs in...
 6.6.69: Does the graph of the learning curve shown in Exercises 5760 indica...
 6.6.70: x 2 + x 4 = 6; [5, 10, 1] by [5, 10, 1]
 6.6.71: 50 x = 2x; [10, 10, 1] by [20, 20, 2]
 6.6.72: x + 6 x = 5; [10, 10, 1] by [7, 10, 1]
 6.6.73: 2 x = x + 1; [5, 5, 1] by [5, 5, 1]
 6.6.74: 3 x  x + 21 3x = 5 3 ; [5, 5, 1] by [5, 5, 1]
 6.6.75: I must have made an error if a rational equation produces no solution.
 6.6.76: I added two rational expressions and found the solution set.
 6.6.77: I can solve the equation 40 x = 15 x  20 by multiplying both sides...
 6.6.78: Im solving a rational equation that became a quadratic equation, so...
 6.6.79: Once a rational equation is cleared of fractions, all solutions of ...
 6.6.80: We find 4 x  2 x + 1 by multiplying each term by the LCD, x(x + 1)...
 6.6.81: All real numbers satisfy the equation 7 x  2 x = 5 x .
 6.6.82: In order to find a number to add to the numerator and denominator o...
 6.6.83: Solve: a 1 x + 1 + x 1  x b , a x x + 1  1 x  1 b = 1.
 6.6.84: Solve: ` x + 1 x + 8 ` = 2 3 .
 6.6.85: Write an original rational equation that has no solution.
 6.6.86: Solve a 4 x  1 b 2 + 2a 4 x  1 b + 1 = 0 by introducing the subst...
 6.6.87: Graph the solution set: e x + 2y 2 x  y 4. (Section 4.4, Example 5)
 6.6.88: Solve: x  4 2  1 5 = 7x + 1 20 . (Section 1.4, Example 4)
 6.6.89: Solve for F: C = 5F  160 9 . (Section 1.5, Example 8)
 6.6.90: Solve for p: qf + pf = pq.
 6.6.91: Solve: 40 x + 40 x + 30 = 2.
 6.6.92: A plane flies at an average rate of 450 miles per hour. It can trav...
Solutions for Chapter 6.6: Rational Equations
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.6: Rational Equations
Get Full SolutionsIntermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 92 problems in chapter 6.6: Rational Equations have been answered, more than 9295 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Chapter 6.6: Rational Equations includes 92 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

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

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 .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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 •

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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 ().

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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