- 2.7.1: Graph each inequality.
- 2.7.2: Graph each inequality.
- 2.7.3: Graph each inequality.
- 2.7.4: Graph each inequality.
- 2.7.5: Graph each inequality.
- 2.7.6: Graph each inequality.
- 2.7.7: Write an inequality to represent the situation, where c is the numb...
- 2.7.8: Graph the inequality.
- 2.7.9: Can she buy 2 CDs and 3 DVDs? Explain
- 2.7.10: Graph each inequality.
- 2.7.11: Graph each inequality.
- 2.7.12: Graph each inequality.
- 2.7.13: Graph each inequality.
- 2.7.14: Graph each inequality.
- 2.7.15: Graph each inequality.
- 2.7.16: Graph each inequality.
- 2.7.17: Graph each inequality.
- 2.7.18: Graph each inequality.
- 2.7.19: Graph each inequality.
- 2.7.20: Graph each inequality.
- 2.7.21: Graph each inequality.
- 2.7.22: The inequality 0.4x + 0.6y 90 represents this situation, where x is...
- 2.7.23: Refer to the graph. If she scores 85 on the midterm and 95 on the f...
- 2.7.24: Write an inequality to represent this situation.
- 2.7.25: Graph the inequality.
- 2.7.26: Will he make enough from 3000 shares of each company?
- 2.7.27: Graph all the points on the coordinate plane to the left of the gra...
- 2.7.28: Graph all the points on the coordinate plane below the graph of y =...
- 2.7.29: Graph each inequality
- 2.7.30: Graph each inequality
- 2.7.31: Graph each inequality
- 2.7.32: Graph each inequality
- 2.7.33: Graph each inequality
- 2.7.34: Graph each inequality
- 2.7.35: Graph each inequality
- 2.7.36: Graph each inequality
- 2.7.37: Graph each inequality
- 2.7.38: Graph each inequality
- 2.7.39: Explain how to determine which region to shade when graphing an ine...
- 2.7.40: Graph y < x
- 2.7.41: Use the information on page 102 to write an inequality that defines...
- 2.7.42: Which could be the inequality for the graph? A y < 3x + 2 B y 3x + ...
- 2.7.43: hat is the solution set of the inequality? 6 - x + 7 -2 F -15 x + 1...
- 2.7.44: Graph each function. Identify the domain and range.
- 2.7.45: Graph each function. Identify the domain and range.
- 2.7.46: Graph each function. Identify the domain and range.
- 2.7.47: Draw a scatter plot and describe the correlation
- 2.7.48: Find a prediction equation.
- 2.7.49: Predict the salary for a representative with 9 years of experience
- 2.7.50: Solve each equation. Check your solution.
- 2.7.51: Solve each equation. Check your solution.
- 2.7.52: Solve each equation. Check your solution.
Solutions for Chapter 2.7: Graphing Inequalities
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
Upper triangular systems are solved in reverse order Xn to Xl.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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).
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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 •
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.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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.