- 5.5.1: Tell what operation on the first inequality gives the second one, a...
- 5.5.2: Find three values of the variable that satisfy each inequality. a. ...
- 5.5.3: Give the inequality graphed on each number line. a. b. c. d. e.
- 5.5.4: Translate each phrase into symbols. a. 3 is more than x b. y is at ...
- 5.5.5: Solve each equation for y. a. 3x + 4y = 5.2 b. 3(y 5) = 2x
- 5.5.6: Solve each inequality and show your work. a. 4.1 + 3.2x > 18 b. 7.2...
- 5.5.7: Solve each inequality and graph the solutions on a number line. a. ...
- 5.5.8: Ezra received $50 from his grandparents for his birthday. He makes ...
- 5.5.9: For each graph, tell what operation moves the two points in the ine...
- 5.5.10: Tell whether each inequality is true or false for the given value. ...
- 5.5.11: Solve each inequality. Explain the meaning of the result. On a numb...
- 5.5.12: Data collected by a motion sensor will vary slightly in accuracy. A...
- 5.5.13: You read the inequality symbols, <, , >, and , as is less than, is ...
- 5.5.14: The table gives equations that model the three vehicles distances i...
- 5.5.15: In Example B, the inequality 8 0.25x < 5 was written to represent t...
- 5.5.16: List the order in which you would perform these operations to get t...
- 5.5.17: The table shows the 2004 U.S. postal rates for letters, large envel...
- 5.5.18: Use the distributive property to rewrite each expression without us...
Solutions for Chapter 5.5: Inequalities in One Variable
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
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.
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 n-l - An).
cond(A) = c(A) = IIAIlIIA-III = 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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > 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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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