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 1.2: Graph each equation in Exercises 14. Let x   3,  2, 1. 0, 1. 2...
 1.3: Graph each equation in Exercises 14. Let x   3,  2, 1. 0, 1. 2...
 1.4: Graph each equation in Exercises 14. Let x   3,  2, 1. 0, 1. 2...
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 1.6: In Exercises 68, use the graph and detennin~ the x4 intercepts if ...
 1.7: In Exercises 68, use the graph and detennin~ the x4 intercepts if ...
 1.8: In Exercises 68, use the graph and detennin~ the x4 intercepts if ...
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 1.27: In Exercises 1535, solve each equation. Then .state whether the eq...
 1.28: In Exercises 1535, solve each equation. Then .state whether the eq...
 1.29: In Exercises 1535, solve each equation. Then .state whether the eq...
 1.30: In Exercises 1535, solve each equation. Then .state whether the eq...
 1.31: In Exercises 1535, solve each equation. Then .state whether the eq...
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 1.33: In Exercises 1535, solve each equation. Then .state whether the eq...
 1.34: In Exercises 1535, solve each equation. Then .state whether the eq...
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 1.36: In Exercises 3643. use the fi.vestep strategy for solvhtg word pr...
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 1.42: In Exercises 3643. use the fi.vestep strategy for solvhtg word pr...
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 1.45: In Exercises 4547. solve each formula for the specified variable. ...
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 1.48: In Exercises 4857, perform the indicated operations and write the ...
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 1.58: Solve each equation in Exercises 5859 by factoring. 2' + 15x  8
 1.59: Solve each equation in Exercises 5859 by factoring. 5x2 + 20x  0
 1.60: Solve each equation in Exercises 6063 by the square root properly....
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 1.64: In Exercises 6465, determine the constant that should be added to ...
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 1.66: Solve each equation in Exercises 6667 by completing the squnre. x1...
 1.67: Solve each equation in Exercises 6667 by completing the squnre. 3x...
 1.68: Solve each equation in Exercises 6870 using the quadratic fomwla. ...
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 1.72: In Exercises 71 72, withow solving the g;ven quadratic equation, d...
 1.73: Solve each equation in Exercises 7381 by the method of your choice...
 1.74: Solve each equation in Exercises 7381 by the method of your choice...
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 1.79: Solve each equation in Exercises 7381 by the method of your choice...
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 1.81: Solve each equation in Exercises 7381 by the method of your choice...
 1.82: The formula W  312 models the weight of a huma n fetus, W, in gra ...
 1.83: As gas prices surge, more Ame ricans are cycling as a way to save m...
 1.84: An architect is allowed 15 square yards of floor space to add a sma...
 1.85: A building casts a shadow that is double the height of the building...
 1.86: Solve each polynomial equation in Exercises 8687. 2x"  50x2
 1.87: Solve each polynomial equation in Exercises 8687. 2x3  x2  18x +...
 1.88: Solve each radical equation hJ Exercises 8889. ~+x 3
 1.89: Solve each radical equation hJ Exercises 8889. ~+Vx+l 5
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 1.91: Solve the equations with rational exponents in Exercises 9091. (x ...
 1.92: Solve each equation in Exercises 9293 by making an appropriate sub...
 1.93: Solve each equation in Exercises 9293 by making an appropriate sub...
 1.94: Solve the equations comaining absolute value in Exercises 9495. l2...
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 1.96: Solve each equation in tercises 96102 by lhe method of your choicf...
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 1.122: In Exercises 122123, use interval notation 10 represent all values...
 1.123: In Exercises 122123, use interval notation 10 represent all values...
 1.124: A car rental agency rents a certain car for $40 per day with unlimi...
 1.125: To receive a B in a course, you must have an average of a t least 8...
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Solutions for Chapter 1: Equations and Inequalities
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 1: Equations and Inequalities
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 126 problems in chapter 1: Equations and Inequalities have been answered, more than 38856 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321782281. This textbook survival guide was created for the textbook: College Algebra , edition: 6. Chapter 1: Equations and Inequalities includes 126 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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.