 65.1: Solve each open sentence. Then graph the solution set.
 65.2: Solve each open sentence. Then graph the solution set.
 65.3: Solve each open sentence. Then graph the solution set.
 65.4: Write an open sentence involving absolute value for the graph.
 65.5: Graph each function.
 65.6: Graph each function.
 65.7: Solve each open sentence. Then graph the solution set.
 65.8: Solve each open sentence. Then graph the solution set.
 65.9: Solve each open sentence. Then graph the solution set.
 65.10: Solve each open sentence. Then graph the solution set.
 65.11: Solve each open sentence. Then graph the solution set.
 65.12: Solve each open sentence. Then graph the solution set.
 65.13: Solve each open sentence. Then graph the solution set.
 65.14: Solve each open sentence. Then graph the solution set.
 65.15: Solve each open sentence. Then graph the solution set.
 65.16: Solve each open sentence. Then graph the solution set.
 65.17: Solve each open sentence. Then graph the solution set.
 65.18: Solve each open sentence. Then graph the solution set.
 65.19: Write an open sentence involving absolute value for each graph.
 65.20: Write an open sentence involving absolute value for each graph.
 65.21: Write an open sentence involving absolute value for each graph.
 65.22: Write an open sentence involving absolute value for each graph.
 65.23: Graph each function.
 65.24: Graph each function.
 65.25: Graph each function.
 65.26: Graph each function.
 65.27: Graph each function.
 65.28: Graph each function.
 65.29: Graph each function.
 65.30: Graph each function.
 65.31: Solve for x.
 65.32: Solve for x.
 65.33: Determine the domain and range for each absolute value function.
 65.34: Determine the domain and range for each absolute value function.
 65.35: Determine the domain and range for each absolute value function.
 65.36: Determine the domain and range for each absolute value function.
 65.37: ANALYZE GRAPHS The circle graph at the right shows the results of a...
 65.38: PHYSICS As part of a physics lab, Tiffany and Curtis determined tha...
 65.39: PING PONG For Exercises 39 and 40, use the following information. E...
 65.40: PING PONG For Exercises 39 and 40, use the following information. E...
 65.41: OPEN ENDED Describe a realworld situation that could be represente...
 65.42: REASONING Determine whether the following statements are sometimes,...
 65.43: REASONING Determine whether the following statements are sometimes,...
 65.44: REASONING Determine whether the following statements are sometimes,...
 65.45: REASONING Determine whether the following statements are sometimes,...
 65.46: CHALLENGE Use the sentence x = 3 1.2. a. Describe the values of x. ...
 65.47: FIND THE ERROR Leslie and Holly are solving x + 3 = 2. Who is corre...
 65.48: Writing in Math Use the data about the technology survey on page 32...
 65.49: Assume n is an integer and solve for n. 2n  3 = 5 A {4, 1} C {1...
 65.50: If p is an integer, which of the following is the solution set for ...
 65.51: REVIEW What is the measure of 1 in the figure shown below? A 83 B 8...
 65.52: FITNESS To achieve the maximum benefits from aerobic activity, your...
 65.53: Solve each inequality. Check your solution.
 65.54: Solve each inequality. Check your solution.
 65.55: Solve each inequality. Check your solution.
 65.56: Find the slope and yintercept of each equation.
 65.57: Find the slope and yintercept of each equation.
 65.58: Find the slope and yintercept of each equation.
 65.59: Express the relation shown in each mapping as a set of ordered pair...
 65.60: Express the relation shown in each mapping as a set of ordered pair...
 65.61: Express the relation shown in each mapping as a set of ordered pair...
 65.62: Solve each equation for the variable specified.
 65.63: Solve each equation for the variable specified.
 65.64: Solve each equation for the variable specified.
 65.65: PREREQUISITE SKILL Solve each inequality. Check your solution.
 65.66: PREREQUISITE SKILL Solve each inequality. Check your solution.
 65.67: PREREQUISITE SKILL Solve each inequality. Check your solution.
 65.68: PREREQUISITE SKILL Solve each inequality. Check your solution.
 65.69: PREREQUISITE SKILL Solve each inequality. Check your solution.
 65.70: PREREQUISITE SKILL Solve each inequality. Check your solution.
Solutions for Chapter 65: Solving Open Sentences Involving Absolute Value
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 65: Solving Open Sentences Involving Absolute Value
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 65: Solving Open Sentences Involving Absolute Value includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Since 70 problems in chapter 65: Solving Open Sentences Involving Absolute Value have been answered, more than 37155 students have viewed full stepbystep solutions from this chapter.

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.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

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.