 P.1.1: A real number is _______ when it can be written as the ratio p q of...
 P.1.2: _______ numbers have infinite nonrepeating decimal representations.
 P.1.3: A _______ number is an integer with exactly two positive factors: i...
 P.1.4: An algebraic expression is a combination of letters called _______ ...
 P.1.5: The _______ of an algebraic expression are those parts separated by...
 P.1.6: Is 5 2 = 2 5?
 P.1.7: Commutative Property of Addition (a) a 1 = a
 P.1.8: Associative Property of Multiplication (b) a(b + c) = ab + ac
 P.1.9: Additive Inverse Property (c) a + b = b + a
 P.1.10: Distributive Property (d) (ab)c = a(bc)
 P.1.11: Associative Property of Addition (e) a + (a) = 0
 P.1.12: Multiplicative Identity Property (f) (a + b) + c = a + (b + c)
 P.1.13: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.14: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.15: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.16: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.17: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.18: In Exercises 1318, determine which numbers are (a) natural numbers,...
 P.1.19: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.20: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.21: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.22: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.23: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.24: In Exercises 1924, use a calculator to find the decimal form of the...
 P.1.25: In Exercises 2528, use a calculator to rewrite the rational number ...
 P.1.26: In Exercises 2528, use a calculator to rewrite the rational number ...
 P.1.27: In Exercises 2528, use a calculator to rewrite the rational number ...
 P.1.28: In Exercises 2528, use a calculator to rewrite the rational number ...
 P.1.29: In Exercises 29 and 30, approximate the numbers and place the corre...
 P.1.30: In Exercises 29 and 30, approximate the numbers and place the corre...
 P.1.31: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.32: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.33: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.34: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.35: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.36: In Exercises 3136, plot the two real numbers on the real number lin...
 P.1.37: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.38: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.39: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.40: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.41: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.42: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.43: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.44: In Exercises 3744, (a) verbally describe the subset of real numbers...
 P.1.45: In Exercises 4552, use inequality and interval notation to represen...
 P.1.46: In Exercises 4552, use inequality and interval notation to represen...
 P.1.47: In Exercises 4552, use inequality and interval notation to represen...
 P.1.48: In Exercises 4552, use inequality and interval notation to represen...
 P.1.49: In Exercises 4552, use inequality and interval notation to represen...
 P.1.50: In Exercises 4552, use inequality and interval notation to represen...
 P.1.51: In Exercises 4552, use inequality and interval notation to represen...
 P.1.52: In Exercises 4552, use inequality and interval notation to represen...
 P.1.53: In Exercises 5356, use interval notation to describe the graph. 4 2...
 P.1.54: In Exercises 5356, use interval notation to describe the graph. 4 2...
 P.1.55: In Exercises 5356, use interval notation to describe the graph. . a...
 P.1.56: In Exercises 5356, use interval notation to describe the graph. .c ...
 P.1.57: In Exercises 5760, give a verbal description of the interval. (6, )
 P.1.58: In Exercises 5760, give a verbal description of the interval. (, 4]
 P.1.59: In Exercises 5760, give a verbal description of the interval. (, 2]
 P.1.60: In Exercises 5760, give a verbal description of the interval. (1, )
 P.1.61: In Exercises 61 66, evaluate the expression. 10
 P.1.62: In Exercises 61 66, evaluate the expression. 0
 P.1.63: In Exercises 61 66, evaluate the expression. 3 3
 P.1.64: In Exercises 61 66, evaluate the expression. 1 2
 P.1.65: In Exercises 61 66, evaluate the expression. 5 5
 P.1.66: In Exercises 61 66, evaluate the expression. 33
 P.1.67: n Exercises 67 and 68, evaluate the expression for the given interv...
 P.1.68: n Exercises 67 and 68, evaluate the expression for the given interv...
 P.1.69: In Exercises 6972, evaluate the expression for the given values of ...
 P.1.70: In Exercises 6972, evaluate the expression for the given values of ...
 P.1.71: In Exercises 6972, evaluate the expression for the given values of ...
 P.1.72: In Exercises 6972, evaluate the expression for the given values of ...
 P.1.73: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.74: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.75: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.76: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.77: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.78: In Exercises 7378, place the correct symbol (<, >, or =) between th...
 P.1.79: In Exercises 7984, find the distance between a and b a = 126, b = 75
 P.1.80: In Exercises 7984, find the distance between a and b a = 126, b = 75
 P.1.81: In Exercises 7984, find the distance between a and b a = 5 2, b = 9 2
 P.1.82: In Exercises 7984, find the distance between a and b a = 1 4, b = 11
 P.1.83: In Exercises 7984, find the distance between a and b a = 16 5 , b =...
 P.1.84: In Exercises 7984, find the distance between a and b a = 7 3, b = 15 8
 P.1.85: The distance between x and 5 is no more than 3.
 P.1.86: The distance between x and 10 is at least 6.
 P.1.87: y is at least six units from 0
 P.1.88: y is at most three units from a.
 P.1.89: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.90: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.91: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.92: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.93: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.94: In Exercises 89 94, use the bar graph, which shows the receipts of ...
 P.1.95: In Exercises 9598, the accounting department of a company is checki...
 P.1.96: In Exercises 9598, the accounting department of a company is checki...
 P.1.97: In Exercises 9598, the accounting department of a company is checki...
 P.1.98: In Exercises 9598, the accounting department of a company is checki...
 P.1.99: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.100: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.101: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.102: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.103: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.104: In Exercises 99104, identify the terms and coefficients of the expr...
 P.1.105: In Exercises 105108, evaluate the expression for each value of x. (...
 P.1.106: In Exercises 105108, evaluate the expression for each value of x. (...
 P.1.107: In Exercises 105108, evaluate the expression for each value of x. (...
 P.1.108: In Exercises 105108, evaluate the expression for each value of x. (...
 P.1.109: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.110: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.111: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.112: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.113: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.114: In Exercises 109114, identify the rule(s) of algebra illustrated by...
 P.1.115: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.116: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.117: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.118: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.119: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.120: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.121: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.122: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.123: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.124: In Exercises 115124, perform the operation(s). (Write fractional an...
 P.1.125: While traveling on the Pennsylvania Turnpike, you pass milepost 57 ...
 P.1.126: The temperature in Bismarck, North Dakota, was 60F at noon, then 23...
 P.1.127: In Exercises 127129, determine whether the statement is true or fal...
 P.1.128: In Exercises 127129, determine whether the statement is true or fal...
 P.1.129: In Exercises 127129, determine whether the statement is true or fal...
 P.1.130: Use a calculator to complete the table. n 1 0.5 0.01 0.0001 0.00000...
 P.1.131: The real numbers A, B, and C are shown on the number line. Determin...
 P.1.132: Match each description with its graph. Which types of real numbers ...
 P.1.133: Describe the real number values of u and v for which u + v is great...
 P.1.134: For what real numbers a is a = a? Explain.
Solutions for Chapter P.1: Prerequisites
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter P.1: Prerequisites
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Chapter P.1: Prerequisites includes 134 full stepbystep solutions. Since 134 problems in chapter P.1: Prerequisites have been answered, more than 58842 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.