 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
ISBN: 9781559537636
Solutions for Chapter 5.5: Inequalities in One Variable
Get Full SolutionsThis textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Since 18 problems in chapter 5.5: Inequalities in One Variable have been answered, more than 2874 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Algebra: An Investigative Approach was written by Patricia and is associated to the ISBN: 9781559537636. Chapter 5.5: Inequalities in One Variable includes 18 full stepbystep solutions.

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.

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).

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
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+ (MoorePenrose 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)jI 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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