 Chapter 1.1: 20 0.16
 Chapter 1.2: 12.2 + (8.45)
 Chapter 1.3: 1 4  _2 3
 Chapter 1.4: 3 5 + (6)
 Chapter 1.5: 7 _1 2 + 5 _1 3
 Chapter 1.6: 11 _5 8  (4 _3 7)
 Chapter 1.7: (0.15)(3.2)
 Chapter 1.8: 2 (0.4)
 Chapter 1.9: 4 _3 2
 Chapter 1.10: (_5 4)( _3 10)
 Chapter 1.11: (2 _3 4)(3 _1 5)
 Chapter 1.12: 7 _1 8 (2)
 Chapter 1.13: LUNCH Angela has $11.56. She spends $4.25 on lunch. How much money ...
 Chapter 1.14: 2
 Chapter 1.15: 5 3
 Chapter 1.16: (7)
 Chapter 1.17: (1) 3
 Chapter 1.18: (0.8) 2
 Chapter 1.19: (1.2) 2
 Chapter 1.20: (_2 3) 2
 Chapter 1.21: (_5 9) 2
 Chapter 1.22: ( _4 11) 2
 Chapter 1.23: GENEALOGY In a family tree, you are generation now. One generation ...
 Chapter 1.24: 5 < 7
 Chapter 1.25: 6 > 8
 Chapter 1.26: 2 2
 Chapter 1.27: 3 3.01
 Chapter 1.28: 1 < 2
 Chapter 1.29: 1 5 < _1 8
 Chapter 1.30: 2 5 _16 40
 Chapter 1.31: 3 4 > 0.8
 Chapter 1.32: n 4 + _ n 3 = _1 2
 Chapter 1.33: 5y + 4 = 2(y  4)
 Chapter 1.34: MONEY If Tabitha has 98 cents and you know she has 2 quarters, 1 di...
 Chapter 1.35: Ax + By = C for x
 Chapter 1.36: a  4b _2 2c = d for a
 Chapter 1.37: A = p + prt for p
 Chapter 1.38: d = b2  4ac for c
 Chapter 1.39: GEOMETRY Alex wants to find the radius of the circular base of a co...
 Chapter 1.40: x + 11 = 42
 Chapter 1.41: 3 x + 6 = 36
 Chapter 1.42: 4x  5 = 25
 Chapter 1.43: x + 7 = 3x  5
 Chapter 1.44: y  5  2 = 10
 Chapter 1.45: 4 3x + 4 = 4x + 8
 Chapter 1.46: BIKING Palomas training goal is to ride four miles on her bicycle i...
 Chapter 1.47: 7w > 28
 Chapter 1.48: 3x + 4 19
 Chapter 1.49: _n 12 + 5 7
 Chapter 1.50: 3(6  5a) < 12a 3
 Chapter 1.51: 2  3z 7(8  2z) + 12
 Chapter 1.52: 8(2x  1) > 11x  17
 Chapter 1.53: PIZZA A group has $75 to order 6 large pizzas each with the same am...
 Chapter 1.54: 4x + 3 < 11 or 2x  1 > 9
 Chapter 1.55: 1 < 3a + 2 < 14
 Chapter 1.56: 1 < 3(d  2) 9
 Chapter 1.57: 5y  4 > 16 or 3y + 2 < 1
 Chapter 1.58: x + 1 > 12
 Chapter 1.59: 2y  9 27
 Chapter 1.60: 5n  8 > 4
 Chapter 1.61: 3b + 11 > 1
 Chapter 1.62: FENCING Don is building a fence around a rectangular plot and wants...
Solutions for Chapter Chapter 1: Equations and Inequalities
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter Chapter 1: Equations and Inequalities
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter Chapter 1: Equations and Inequalities includes 62 full stepbystep solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Since 62 problems in chapter Chapter 1: Equations and Inequalities have been answered, more than 53487 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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

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

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

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

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·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.