 2.6.1: Given , determine the inequality obtained if (a) 5 is added to both...
 2.6.2: Given , determine the inequality obtained if (a) 7 is added to both...
 2.6.3: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.4: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.5: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.6: Exer. 312: Express the inequality as an interval, and sketch its gr...
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 2.6.8: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.9: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.10: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.11: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.12: Exer. 312: Express the inequality as an interval, and sketch its gr...
 2.6.13: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.14: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.15: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.16: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.17: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.18: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.19: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.20: Exer. 1320: Express the interval as an inequality in the variable x.
 2.6.21: Exer. 2170: Solve the inequality, and express the solutions in term...
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 2.6.69: Exer. 2170: Solve the inequality, and express the solutions in term...
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 2.6.71: Exer. 7172: Solve part (a) and use that answer to determine the ans...
 2.6.72: Exer. 7172: Solve part (a) and use that answer to determine the ans...
 2.6.73: Exer. 7376: Express the statement in terms of an inequality involvi...
 2.6.74: Exer. 7376: Express the statement in terms of an inequality involvi...
 2.6.75: Exer. 7376: Express the statement in terms of an inequality involvi...
 2.6.76: Exer. 7376: Express the statement in terms of an inequality involvi...
 2.6.77: Temperature readings on the Fahrenheit and Celsius scales are relat...
 2.6.78: According to Hookes law, the force F (in pounds) required to stretc...
 2.6.79: Ohms law in electrical theory states that if R denotes the resistan...
 2.6.80: If two resistors and are connected in parallel in an electrical cir...
 2.6.81: Shown in the figure is a simple magnifier consisting of a convex le...
 2.6.82: To treat arrhythmia (irregular heartbeat), a drug is fed intravenou...
 2.6.83: A construction firm is trying to decide which of two models of a cr...
 2.6.84: A consumer is trying to decide whether to purchase car A or car B. ...
 2.6.85: A persons height will typically decrease by 0.024 inch each year af...
Solutions for Chapter 2.6: Inequalities
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 2.6: Inequalities
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.6: Inequalities includes 85 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Since 85 problems in chapter 2.6: Inequalities have been answered, more than 37620 students have viewed full stepbystep solutions from this chapter.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.