 6.6.6.1.308: Two or more expressions joined by an inequality sign is called a(n) .
 6.6.6.1.309: a) a 6 b means that a is than b. b) a 7 b means that a is than b.
 6.6.6.1.310: An inequality of the form a 6 x 6 b is called a(n) inequality.
 6.6.6.1.311: When solving an inequality, if you obtain a statement that is alway...
 6.6.6.1.312: When solving an inequality, if you obtain a statement that is alway...
 6.6.6.1.313: When both sides of an inequality are multiplied or divided by a(n) ...
 6.6.6.1.314: To indicate the solution set of x < 5 on the number line, we draw a...
 6.6.6.1.315: To indicate the solution set of x 5 on the number line, we draw a(n...
 6.6.6.1.316: In Exercises 926, graph the solution set of the inequality, where x...
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 6.6.6.1.331: In Exercises 926, graph the solution set of the inequality, where x...
 6.6.6.1.332: In Exercises 926, graph the solution set of the inequality, where x...
 6.6.6.1.333: In Exercises 926, graph the solution set of the inequality, where x...
 6.6.6.1.334: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.335: In Exercises 2746, graph the solution set of the inequality, where ...
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 6.6.6.1.342: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.343: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.344: In Exercises 2746, graph the solution set of the inequality, where ...
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 6.6.6.1.351: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.352: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.353: In Exercises 2746, graph the solution set of the inequality, where ...
 6.6.6.1.354: Indianapolis 500 The following bar graph shows the average speed, i...
 6.6.6.1.355: Unemployment Rate The following graph shows the unemployment rate i...
 6.6.6.1.356: DVD Rental Movie Mania offers two rental plans. One has an annual f...
 6.6.6.1.357: Salary Plans Bobby Exler recently accepted a sales position in Port...
 6.6.6.1.358: College Employment To be eligible for financial assistance for coll...
 6.6.6.1.359: Anniversary Patricia and Joe Kuby are celebrating their 75th weddin...
 6.6.6.1.360: Parking Costs A parking garage in Richmond, Virginia, charges $1.75...
 6.6.6.1.361: Making a Profit For a business to realize a profit, its revenue, R,...
 6.6.6.1.362: Finding Velocity The velocity, v, in feet per second, t sec after a...
 6.6.6.1.363: Speed Limit The minimum speed for vehicles on a highway is 40 mph, ...
 6.6.6.1.364: A Grade of B In Example 9 on page 332, what range of grades on the ...
 6.6.6.1.365: Tent Rental The Cuhayoga Community College Planning Committee wants...
 6.6.6.1.366: Painting a House Donovan Davis is painting the exterior of his hous...
 6.6.6.1.367: Final Exam Teresas five test grades for the semester are 86%, 78%, ...
 6.6.6.1.368: A student multiplied both sides of the inequality 1 3 x 4 by 3 an...
 6.6.6.1.369: Find a newspaper or a magazine article that contains the mathematic...
Solutions for Chapter 6.6: Algebra, Graphs, and Functions
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 6.6: Algebra, Graphs, and Functions
Get Full SolutionsSince 62 problems in chapter 6.6: Algebra, Graphs, and Functions have been answered, more than 74922 students have viewed full stepbystep solutions from this chapter. Chapter 6.6: Algebra, Graphs, and Functions includes 62 full stepbystep solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665.

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.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·