 1.2.1.2.1: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.2: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.3: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.4: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.5: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.6: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.7: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.8: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.9: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.10: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.11: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.12: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.13: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.14: Find the value of each algebraic expression at the given replacemen...
 1.2.1.2.15: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.16: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.17: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.18: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.19: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.20: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.21: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.22: Write each set in roster form. (List the elements of each set.) See...
 1.2.1.2.23: Graph each set on a number line.50, 2, 4, 66
 1.2.1.2.24: Graph each set on a number line.51, 3, 5, 76
 1.2.1.2.25: Graph each set on a number line.e12,23 f
 1.2.1.2.26: Graph each set on a number line.e14,13 f
 1.2.1.2.27: Graph each set on a number line.5 2, 6, 106
 1.2.1.2.28: Graph each set on a number line.5 1, 2, 36
 1.2.1.2.29: Graph each set on a number line.e 13 , 113 f
 1.2.1.2.30: Graph each set on a number line.e  12 , 112 f
 1.2.1.2.31: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.32: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.33: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.34: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.35: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.36: List the elements of the set e 3, 0, 27, 236, 2 5 , 134 f that are...
 1.2.1.2.37: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.38: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.39: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.40: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.41: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.42: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.43: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.44: Place or in the space provided to make each statement true. See Exa...
 1.2.1.2.45: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.46: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.47: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.48: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.49: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.50: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.51: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.52: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.53: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.54: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.55: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.56: Determine whether each statement is true or false. See Examples 4an...
 1.2.1.2.57: Find each absolute value. See Example 6. 0 2
 1.2.1.2.58: Find each absolute value. See Example 6.8 0
 1.2.1.2.59: Find each absolute value. See Example 6.4 0
 1.2.1.2.60: Find each absolute value. See Example 6.6 0
 1.2.1.2.61: Find each absolute value. See Example 6.0
 1.2.1.2.62: Find each absolute value. See Example 6.1 0
 1.2.1.2.63: Find each absolute value. See Example 6. 0 3 0
 1.2.1.2.64: Find each absolute value. See Example 6. 0 11 0
 1.2.1.2.65: Write the opposite of each number. See Example 7.6.2
 1.2.1.2.66: Write the opposite of each number. See Example 7.7.8
 1.2.1.2.67: Write the opposite of each number. See Example 7.47
 1.2.1.2.68: Write the opposite of each number. See Example 7.95
 1.2.1.2.69: Write the opposite of each number. See Example 7. 23
 1.2.1.2.70: Write the opposite of each number. See Example 7. 143
 1.2.1.2.71: Write the opposite of each number. See Example 7.0
 1.2.1.2.72: Write the opposite of each number. See Example 7.10.3
 1.2.1.2.73: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.74: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.75: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.76: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.77: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.78: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.79: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.80: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.81: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.82: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.83: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.84: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.85: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.86: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.87: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.88: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.89: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.90: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.91: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.92: Translating Write each phrase as an algebraic expression. Use the v...
 1.2.1.2.93: Use the bar graph below to complete the given table by estimating t...
 1.2.1.2.94: Use the bar graph below to complete the given table by estimating t...
 1.2.1.2.95: Use the bar graph below to complete the given table by estimating t...
 1.2.1.2.96: Use the bar graph below to complete the given table by estimating t...
 1.2.1.2.97: Explain why 1 22 and  0 2 0 simplify to different numbers.
 1.2.1.2.98: The boxed definition of absolute value states that 0 a 0 = a if a ...
 1.2.1.2.99: In your own words, explain why every natural number is also a ratio...
 1.2.1.2.100: In your own words, explain why every irrational number is a real nu...
 1.2.1.2.101: In your own words, explain why the empty set is a subset of every set.
 1.2.1.2.102: In your own words, explain why every set is a subset of itself.
Solutions for Chapter 1.2: Algebraic Expressions and Sets of Numbers
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 1.2: Algebraic Expressions and Sets of Numbers
Get Full SolutionsSince 102 problems in chapter 1.2: Algebraic Expressions and Sets of Numbers have been answered, more than 66567 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Chapter 1.2: Algebraic Expressions and Sets of Numbers includes 102 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

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).

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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 •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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