 4.1.1: The addition property of inequality states that if a 6 b, then a + c .
 4.1.2: The positive multiplication property of inequality states that if a...
 4.1.3: The negative multiplication property of inequality states that if a...
 4.1.4: The linear inequality 3x  4 7 5 can be solved by first to both si...
 4.1.5: In solving an inequality, if you eliminate the variable and obtain ...
 4.1.6: In solving an inequality, if you eliminate the variable and obtain ...
 4.1.7: The algebraic translation of x is at least 7 is .
 4.1.8: The algebraic translation of x is at most 7 is .
 4.1.9: The algebraic translation of x is no more than 7 is .
 4.1.10: The algebraic translation of x is no less than 7 is .
 4.1.11: 2x  11 6 3(x + 2)
 4.1.12: 4(x + 2) 7 3x + 20
 4.1.13: 1  (x + 3) 4  2x
 4.1.14: 5(3  x) 3x  1
 4.1.15: x 4  1 2 x 2 + 1
 4.1.16: 3x 10 + 1 1 5  x 10
 4.1.17: 1  x 2 7 4
 4.1.18: 7  4 5 x 6 3 5
 4.1.19: x  4 6 x  2 9 + 5 18
 4.1.20: 4x  3 6 + 2 2x  1 12
 4.1.21: 4(3x  2)  3x 6 3(1 + 3x)  7
 4.1.22: 3(x  8)  2(10  x) 6 5(x  1)
 4.1.23: 8(x + 1) 7(x + 5) + x
 4.1.24: 4(x  1) 3(x  2) + x
 4.1.25: 3x 6 3(x  2)
 4.1.26: 5x 6 5(x  3)
 4.1.27: 7(x + 4)  13 6 12 + 13(3 + x)
 4.1.28: 3[7x  (2x  3)] 7 2(x + 1)
 4.1.29: 6  2 3 (3x  12) 2 5 (10x + 50)
 4.1.30: 2 7 (7  21x)  4 7 10  3 11 (11x  11)
 4.1.31: 3[3(x + 5) + 8x + 7] + 5[3(x  6)  2(3x  5)] 6 2(4x + 3)
 4.1.32: 5[3(2  3x)  2(5  x)]  6[5(x  2)  2(4x  3)] 6 3x + 19
 4.1.33: Let f(x) = 3x + 2 and g(x) = 5x  8. Find all values of x for which...
 4.1.34: Let f(x) = 2x  9 and g(x) = 5x + 4. Find all values of x for which...
 4.1.35: Let f(x) = 2 5 (10x + 15) and g(x) = 1 4 (8  12x). Find all values...
 4.1.36: Let f(x) = 3 5 (10x  15) + 9 and g(x) = 3 8 (16  8x)  7. Find al...
 4.1.37: Let f(x) = 1  (x + 3) + 2x. Find all values of x for which f(x) is...
 4.1.38: Let f(x) = 2x  11 + 3(x + 2). Find all values of x for which f(x) ...
 4.1.39: 2(x + 3) 7 6  {4[x  (3x  4)  x] + 4}
 4.1.40: 3(4x  6) 6 4  {5x  [6x  (4x  (3x + 2))]}
 4.1.41: ax + b 7 c; Assume a 6 0.
 4.1.42: ax + b c 7 b; Assume a 7 0 and c 6 0.
 4.1.43: y1 y2
 4.1.44: y1 y2
 4.1.45: y1 6 y2
 4.1.46: y1 7 y2
 4.1.47: Use interval notation to write an inequality that expresses for whi...
 4.1.48: Use interval notation to write an inequality that expresses for whi...
 4.1.49: What is the relationship between passion and intimacy on the interv...
 4.1.50: What is the relationship between intimacy and commitment on the int...
 4.1.51: What is the relationship between passion and commitment on the inte...
 4.1.52: What is the relationship between passion and commitment on the inte...
 4.1.53: What is the maximum level of intensity for passion? After how many ...
 4.1.54: After approximately how many years do levels of intensity for commi...
 4.1.55: The percentage, P, of U.S. voters who use electronic voting systems...
 4.1.56: The percentage, P, of U.S. voters who use punch cards or lever mach...
 4.1.57: Find values of t such that W 6 M. Describe what this means in terms...
 4.1.58: Find values of t such that W 7 M. Describe what this means in terms...
 4.1.59: A truck can be rented from Basic Rental for $50 a day plus $0.20 pe...
 4.1.60: You are choosing between two telephone plans. Plan A has a monthly ...
 4.1.61: A city commission has proposed two tax bills. The first bill requir...
 4.1.62: A local bank charges $8 per month plus 5 per check. The credit unio...
 4.1.63: A company manufactures and sells blank audiocassette tapes. The wee...
 4.1.64: A company manufactures and sells personalized stationery. The weekl...
 4.1.65: An elevator at a construction site has a maximum capacity of 3000 p...
 4.1.66: An elevator at a construction site has a maximum capacity of 2500 p...
 4.1.67: When graphing the solutions of an inequality, what does a parenthes...
 4.1.68: When solving an inequality, when is it necessary to change the sens...
 4.1.69: Describe ways in which solving a linear inequality is similar to so...
 4.1.70: Describe ways in which solving a linear inequality is different fro...
 4.1.71: When solving a linear inequality, describe what happens if the solu...
 4.1.72: When solving a linear inequality, describe what happens if the solu...
 4.1.73: What is the slope of each model in Exercises 5556? What does this m...
 4.1.74: 3(x  6) 7 2x  2
 4.1.75: 2(x + 4) 7 6x + 16
 4.1.76: Use a graphing utilitys TABLE feature to verify your work in Exerci...
 4.1.77: 12x  10 7 2(x  4) + 10x
 4.1.78: 2x + 3 7 3(2x  4)  4x
 4.1.79: A bank offers two checking account plans. Plan A has a base service...
 4.1.80: I began the solution of 5  3(x + 2) 7 10x by simplifying the left ...
 4.1.81: I have trouble remembering when to reverse the direction of an ineq...
 4.1.82: If you tell me that three times a number is less than two times tha...
 4.1.83: Whenever I solve a linear inequality in which the coefficients of t...
 4.1.84: The inequality 3x 7 6 is equivalent to 2 7 x.
 4.1.85: The smallest real number in the solution set of 2x 7 6 is 4.
 4.1.86: If x is at least 7, then x 7 7.
 4.1.87: The inequality 3x 7 6 is equivalent to 2 7 x.
 4.1.88: Find a so that the solution set of ax + 4 12 is [8, ).
 4.1.89: Whats wrong with this argument? Suppose x and y represent two real ...
 4.1.90: If f(x) = x2  2x + 5, find f(4). (Section 2.1, Example 3)
 4.1.91: Solve the system: 2x  y  z = 3 3x  2y  2z = 5 x + y + 2z = 4...
 4.1.92: Simplify: 2x4 y2 4xy3 3 . (Section 1.6, Example 9)
 4.1.93: Consider the sets A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}. a. Writ...
 4.1.94: a. Solve: x  3 6 5. b. Solve: 2x + 4 6 14. c. Give an example of a...
 4.1.95: a. Solve: 2x  6 4. b. Solve: 5x + 2 17. c. Give an example of a n...
Solutions for Chapter 4.1: Solving Linear Inequalities
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 4.1: Solving Linear Inequalities
Get Full SolutionsChapter 4.1: Solving Linear Inequalities includes 95 full stepbystep solutions. Since 95 problems in chapter 4.1: Solving Linear Inequalities have been answered, more than 45737 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

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

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Lucas numbers
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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

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

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.

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

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