 1.6.1: 2 =
 1.6.2: True or False x 0 for any real number x.
 1.6.3: The solution set of the equation isx = 5
 1.6.4: The solution set of the inequalityx 6 5
 1.6.5: True or False The equation has no solution.x = 2
 1.6.6: . True or False The inequality has the set of real numbers as its s...
 1.6.7: In 734,solve each equation. 2x = 6
 1.6.8: In 734,solve each equation. 3x = 12
 1.6.9: In 734,solve each equation. 2x + 3 = 5
 1.6.10: In 734,solve each equation. 3x  1 = 2
 1.6.11: In 734,solve each equation. 1  4t + 8 = 13
 1.6.12: In 734,solve each equation. 1  2z + 6 = 9
 1.6.13: In 734,solve each equation. 2x = 8
 1.6.14: In 734,solve each equation. x = 1
 1.6.15: In 734,solve each equation. 2x = 4
 1.6.16: In 734,solve each equation. 3x = 9
 1.6.17: In 734,solve each equation. 23 x = 9
 1.6.18: In 734,solve each equation. 34 x = 9
 1.6.19: In 734,solve each equation.x 3 + 2 5 ` = 2
 1.6.20: In 734,solve each equation.x 2  1 3 ` = 1
 1.6.21: In 734,solve each equation.u  2 =  1 2
 1.6.22: In 734,solve each equation.2  v = 1
 1.6.23: In 734,solve each equation.4  2x = 3
 1.6.24: In 734,solve each equation.5  ` 1 2 x ` = 3
 1.6.25: In 734,solve each equation.x2  9 = 0
 1.6.26: In 734,solve each equation.x2  16 = 0
 1.6.27: In 734,solve each equation.x2  2x = 3
 1.6.28: In 734,solve each equation.x2 + x = 12
 1.6.29: In 734,solve each equation.x2 + x  1 = 1
 1.6.30: In 734,solve each equation.x2 + 3x  2 = 2
 1.6.31: In 734,solve each equation.3x  2 2x  3 ` = 2
 1.6.32: In 734,solve each equation.2x + 1 3x + 4 ` = 1
 1.6.33: In 734,solve each equation.x2 + 3x = x2  2x
 1.6.34: In 734,solve each equation.x2  2x = x2 + 6x
 1.6.35: In 3562,solve each inequality. Express your answer using set notati...
 1.6.36: In 3562,solve each inequality. Express your answer using set notati...
 1.6.37: In 3562,solve each inequality. Express your answer using set notati...
 1.6.38: In 3562,solve each inequality. Express your answer using set notati...
 1.6.39: In 3562,solve each inequality. Express your answer using set notati...
 1.6.40: In 3562,solve each inequality. Express your answer using set notati...
 1.6.41: In 3562,solve each inequality. Express your answer using set notati...
 1.6.42: In 3562,solve each inequality. Express your answer using set notati...
 1.6.43: In 3562,solve each inequality. Express your answer using set notati...
 1.6.44: In 3562,solve each inequality. Express your answer using set notati...
 1.6.45: In 3562,solve each inequality. Express your answer using set notati...
 1.6.46: In 3562,solve each inequality. Express your answer using set notati...
 1.6.47: In 3562,solve each inequality. Express your answer using set notati...
 1.6.48: In 3562,solve each inequality. Express your answer using set notati...
 1.6.49: In 3562,solve each inequality. Express your answer using set notati...
 1.6.50: In 3562,solve each inequality. Express your answer using set notati...
 1.6.51: In 3562,solve each inequality. Express your answer using set notati...
 1.6.52: In 3562,solve each inequality. Express your answer using set notati...
 1.6.53: In 3562,solve each inequality. Express your answer using set notati...
 1.6.54: In 3562,solve each inequality. Express your answer using set notati...
 1.6.55: In 3562,solve each inequality. Express your answer using set notati...
 1.6.56: In 3562,solve each inequality. Express your answer using set notati...
 1.6.57: In 3562,solve each inequality. Express your answer using set notati...
 1.6.58: In 3562,solve each inequality. Express your answer using set notati...
 1.6.59: In 3562,solve each inequality. Express your answer using set notati...
 1.6.60: In 3562,solve each inequality. Express your answer using set notati...
 1.6.61: In 3562,solve each inequality. Express your answer using set notati...
 1.6.62: In 3562,solve each inequality. Express your answer using set notati...
 1.6.63: Body Temperature Normal human body temperature is 98.6F. If a tempe...
 1.6.64: Household Voltage In the United States, normal household voltage is...
 1.6.65: Reading Books A Gallup poll conducted May 2022, 2005, found that Am...
 1.6.66: Speed of Sound According to data from the Hill Aerospace Museum (Hi...
 1.6.67: Express the fact that x differs from 3 by less than as an inequalit...
 1.6.68: Express the fact that x differs from by less than 1 as an inequalit...
 1.6.69: Express the fact that x differs from by more than 2 as an inequalit...
 1.6.70: Express the fact that x differs from 2 by more than 3 as an inequal...
 1.6.71: In 7176, find a and b. Ifx  1 6 3, a 6 x + 4 6 b.
 1.6.72: In 7176, find a and b. Ifx + 2 6 5, a 6 x  2 6 b.
 1.6.73: In 7176, find a and b. Ifx + 4 2, a 2x  3 b
 1.6.74: In 7176, find a and b. Ifx  3 1, a 3x + 1 b
 1.6.75: In 7176, find a and b. Ifa 1 x  10 x
 1.6.76: In 7176, find a and b. Ifa 1 x + 5 x + 1 3, b.
 1.6.77: Show that: if and , thenb  a = 11b  1a211b + 1a2 a 7
 1.6.78: Show thata a.
 1.6.79: Prove the triangle inequality[Hint: Expand and use the result of 78...
 1.6.80: Prove that [Hint: Apply the triangle inequality from to ]. a  b a ...
 1.6.81: If show that the solution set of the inequality consists of all num...
 1.6.82: If show that the solution set of the inequality consists of all num...
 1.6.83: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.84: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.85: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.86: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.87: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.88: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.89: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.90: In 8390, use the results found in 81 and 82 to solve each inequalit...
 1.6.91: Solve 3x  2x + 1 = 4.
 1.6.92: Solve x + 3x  2 = 2.
 1.6.93: The equation has no solution. Explain whyx = 2
 1.6.94: The inequality has all real numbers as solutions. Explain why.x 7 0.5
 1.6.95: The inequality thathas as solution set Explain why. x 7 0 5x x Z 06
Solutions for Chapter 1.6: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 1.6
Get Full SolutionsSince 95 problems in chapter 1.6 have been answered, more than 61325 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Chapter 1.6 includes 95 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·