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P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 114
114 Chapter P Prerequisites: Fundamental Concepts of Algebra
54. I find the hardest part in solving a word problem is writing 62. A thief steals a number of rare plants from a nursery.On the
the equation that models the verbal conditions. way out, the thief meets three security guards, one after
55. After a 35% reduction, a computer’s price is $780, so I another. To each security guard, the thief is forced to give
determined the original price, x, by solving x - 0.35 = 780. one-half the plants that he still has, plus 2 more. Finally, the
56. When I use the square root property to determine the length thief leaves the nursery with 1 lone palm. How many plants
of a right triangle’s side, I don’t even bother to list the were originally stolen?
negative square root.
57. The perimeter of a plot of land in the shape of a right triangle Group Exercise
is 12 miles. If one leg of the triangle exceeds the other leg by 63. One of the best ways to learn how to solve a word problem in
1 mile,find the length of each boundary of the land. algebra is to design word problems of your own. Creating a
58. The price of a dress is reduced by 40%.When the dress still word problem makes you very aware of precisely how much
does not sell,it is reduced by 40% of the reduced price.If the information is needed to solve the problem. You must also
price of the dress after both reductions is $72, what was the focus on the best way to present information to a reader and
original price? on how much information to give. As you write your
59. In a film, the actor Charles Coburn plays an elderly “uncle” problem, you gain skills that will help you solve problems
character criticized for marrying a woman when he is 3 times created by others.
her age. He wittily replies, “Ah, but in 20 years time I shall The group should design five different word problems
only be twice her age.” How old is the “uncle” and the that can be solved using equations. All of the problems
woman? should be on different topics. For example, the group should
60. Suppose that we agree to pay you 8¢ for every problem in not have more than one problem on the perimeter of a
this chapter that you solve correctly and fine you 5¢ for every rectangle. The group should turn in both the problems and
problem done incorrectly.If at the end of 26 problems we do their algebraic solutions.
not owe each other any money,how many problems did you (If you’re not sure where to begin,consider the graph for
solve correctly? Exercises 5–6 and the data that we did not use regarding
attitudes about women in combat.)
61. It was wartime when the Ricardos found out Mrs. Ricardo
was pregnant. Ricky Ricardo was drafted and made out a
will, deciding that $14,000 in a savings account was to be Preview Exercises
divided between his wife and his child-to-be.Rather strangely, Exercises 64–66 will help you prepare for the material covered in
and certainly with gender bias, Ricky stipulated that if the the next section.
child were a boy, he would get twice the amount of the
mother’s portion.If it were a girl,the mother would get twice 64. Is -1 a solution of 3 - 2x … 11?
the amount the girl was to receive. We’ll never know what 65. Solve: -2x - 4 = x + 5.
Ricky was thinking of, for (as fate would have it) he did not
return from war.Mrs.Ricardo gave birth to twins—a boy and 66. Solve: x + 3 = x - 2 + 1.
a girl. How was the money divided? 4 3 4
Section P.9 Linear Inequalities and Absolute
Objectives Value Inequalities
Use interval notation. ent-a-Heap,a car rental com-
Find intersections and unions Rpany,charges $125 per week
of intervals. plus $0.20 per mile to rent one
Solve linear inequalities. of their cars. Suppose you are
Solve compound inequalities. limited by how much money
Solve absolute value you can spend for the week:
inequalities. You can spend at most $335.
If we let x represent the
number of miles you drive
the heap in a week,we
can write an inequality
that models the given
conditions:
the charge of must be less
The weekly $0.20 per mile than
charge of $125 plus for x miles or equal to $335.
125 + 0.20x 335.
P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 115
Section P.9 Linear Inequalities and Absolute Value Inequalities 115
Placing an inequality symbol between a polynomial of degee 1 and a constant
results in a linear inequality in one variable. In this section, we will study how to
solve linear inequalities such as 125 + 0.20x … 335. Solving an inequality is the
process of finding the set of numbers that make the inequality a true statement.
These numbers are called the solutions of the inequality and we say that they satisfy
the inequality. The set of all solutions is called the solution set of the inequality.
Set-builder notation and a new notation, called interval notation, are used to
represent these solution sets.We begin this section by looking at interval notation.
Use interval notation. Interval Notation
Some sets of real numbers can be represented using interval notation.Suppose that
a and b are two real numbers such that a 6 b.
Interval Notation Graph
The open interval 1a, b2 represents the set of real
numbers between,but not including,a and b. x
a (a, b) b
(a, b)={x | a x b}
The parentheses in the graph
x is greater than a (a < x) and in interval notation indicate
and that a and b, the endpoints, are
x is less than b (x < b). excluded from the interval.
The closed interval 3a, b4 represents the set of real x
numbers between,and including,a and b. a [a, b] b
[a, b]={x | a x b} The square brackets in the graph
x is greater than or equal to a (a ≤ x) and in interval notation indicate
and that a and b, the endpoints, are
x is less than or equal to b (x ≤ b). included in the interval.
The infinite interval 1a, q2 represents the set of
real numbers that are greater than a.
x
a (a, )
(a, )={x | x a}
The infinity symbol does not The parenthesis indicates that a
represent a real number. It is excluded from the interval.
indicates that the interval
extends indefinitely to the right.
The infinite interval 1-q, b4 represents the set of
real numbers that are less than or equal to b. x
(− , b] b
(–, b]={x | x b}
The negative infinity symbol The square bracket indicates
indicates that the interval that b is included in the interval.
extends indefinitely to the left.
Parentheses and Brackets in Interval Notation
Parentheses indicate endpoints that are not included in an interval. Square
brackets indicate endpoints that are included in an interval. Parentheses are
always used with q or -q.
P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 116
116 Chapter P Prerequisites: Fundamental Concepts of Algebra
Table P.7 lists nine possible types of intervals used to describe subsets of real
numbers.
Table P.7 Intervals on the Real Number Line
Let a and b be real numbers such that a
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