Transcript of "Linear Equations and Inequalities : Ch-02 section-2"
1.
2-2 Linear Equations and Inequalities 105
92. Petroleum Consumption. Analyzing data from the
United States Energy Department for the period between
1920 and 1960 reveals that petroleum consumption as a
percentage of all energy consumed (wood, coal, petro-
leum, natural gas, hydro, and nuclear) increased almost
linearly. Percentages for this period are given in the table.
Year
1920
1930
1940
1950
1960
Consumption (%)
11
22
29
37
44
(A) Use regression analysis to ﬁnd a linear regression
function f(x) for this data, where x is the number of
years since 1900.
(B) Use f(x) to estimate (to the nearest one percent) the
percent of petroleum consumption in 1932. In 1956.
(C) If we assume that f(x) continues to provide a good
description of the percentage of petroleum
consumption after 1960, when would this percentage
reach 100%? Is this likely to happen? Explain.
Section 2-2 Linear Equations and Inequalities
Solving Linear Equations
Solving Linear Inequalities
Solving Equations and Inequalities Involving Absolute Value
Application
In this section we discuss methods for solving equations and inequalities that
involve linear functions. Some problems are best solved using algebraic tech-
niques, while others beneﬁt from a graphical approach. Since graphs often give
additional insight into relationships, especially in applications, we will usu-
ally emphasize graphical techniques over algebraic methods. But you must be cer-
tain to master both. There are problems in this section that can only be solved
algebraically. Later we will also encounter problems that can only be solved
graphically.
Solving Linear Equations
In the preceding section we found the x intercept of a linear function f(x) ϭ mx
ϩ b by solving the equation f(x) ϭ 0. Now we want to apply the same ideas to
some more complicated equations.
Solving an Equation Algebraically
Solve 5x Ϫ 8 ϭ 2x ϩ 1 and check.
S o l u t i o n We use the familiar properties of equality to transform the given equation into
an equivalent equation that has an obvious solution (see Section A-8).
Original equation
Add 8 to both sides.5x Ϫ 8 ؉ 8 ϭ 2x ϩ 1 ؉ 8
5x Ϫ 8 ϭ 2x ϩ 1
E X A M P L E
1
2.
106 2 LINEAR AND QUADRATIC FUNCTIONS
Combine like terms.
Subtract 2x from both sides.
Combine like terms.
Divide both sides by 3.
Simplify.
The solution set for this last equation is obvious:
Solution set: {3}
It follows from the properties of equality that {3} is also the solution set of all
the preceding equations in our solution. [Note: If an equation has only one ele-
ment in its solution set, we generally use the last equation (in this case, x ϭ 3)
rather than set notation to represent the solution.]
C h e c k Original equation
Substitute x ϭ 3.
Simplify each side.
A true statement
Solve and check: 2x ϩ 1 ϭ 4x ϩ 5
We can also use a graphing utility to solve equations of this type. From a
graphical viewpoint, a solution to an equation of the form f(x) ϭ g(x) is an inter-
section point of the graphs of f and g. Figure 1 shows a graphical solution to
Example 1 using a built-in intersection routine. Most graphing utilities have such
a routine (consult your owner’s manual or the graphing utility manual for this
text; see the Preface). If yours does not, then use zoom and trace to approximate
intersection points.
Ϫ10
Ϫ10
10
10
y2 ϭ 2x ϩ 1
y1 ϭ 5x Ϫ 8
FIGURE 1
Graphical solution of
5x Ϫ 8 ϭ 2x ϩ 1.
M A T C H E D P R O B L E M
1
7 ⁄ 7
15 Ϫ 8 6 ϩ 1
5(3) Ϫ 8 2(3) ϩ 1
5x Ϫ 8 ϭ 2x ϩ 1
x ϭ 3
3x
3
ϭ
9
3
3x ϭ 9
5x ؊ 2x ϭ 2x ϩ 9 ؊ 2x
5x ϭ 2x ϩ 9
106 2 LINEAR AND QUADRATIC FUNCTIONS106 2 LINEAR AND QUADRATIC FUNCTIONS
3.
2-2 Linear Equations and Inequalities 107
Explore/Discuss
1
An equation that is true for all permissible values of the variable is
called an identity. An equation that is true for some values of the vari-
able and false for others is called a conditional equation. Use algebraic
and/or graphical techniques to solve each of the following and identify
any identities.
(A) 2(x Ϫ 4) ϭ 2x Ϫ 8
(B) 2(x Ϫ 4) ϭ 3x Ϫ 12
(C) 2(x Ϫ 4) ϭ 2x Ϫ 12
Solving an Equation with a Variable in the Denominator
Solve algebraically and graphically:
S o l u t i o n We begin with an algebraic solution. Note that 0 must be excluded from the per-
missible values of x because division by 0 is not permitted. To clear the fractions,
we multiply both sides of the equation by 3(2x) ϭ 6x, the least common denom-
inator (LCD) of all fractions in the equation. (For a discussion of LCDs and how
to ﬁnd them, see Section A-4.)
x 0
The solution set is { }.
Figure 2 shows the graphical solution. Note that to seven dec-
imal places.
Solve algebraically and graphically:
Remark Which solution method should you use—algebraic or graphical? In Example 1,
both the algebraic solution and the graphical solution produced the exact solution,
7
3x
ϩ 2 ϭ
1
x
Ϫ
3
5
M A T C H E D P R O B L E M
2
111
34 Ϸ 3.2647059,
111
34x ϭ
111
34
Ϫ34x ϭ Ϫ111
The equation is now free of
fractions.
21 Ϫ 18x ϭ 16x Ϫ 90
6x ؒ
7
2x
Ϫ 6x ؒ 3 ϭ 6x ؒ
8
3
Ϫ 6x ؒ
15
x
Multiply by 6x, the LCD. This
and the next step usually can
be done mentally.
6x
7
2x
Ϫ 3ϭ 6x
8
3
Ϫ
15
x
7
2x
Ϫ 3 ϭ
8
3
Ϫ
15
x
7
2x
Ϫ 3 ϭ
8
3
Ϫ
15
x
E X A M P L E
2
Ϫ10
Ϫ10
10
10
y2 ϭ
15
x
8
3
Ϫ
y1 ϭ
7
2x
Ϫ 3
FIGURE 2
Graphical solution of
8
3
Ϫ
15
x
.
7
2x
Ϫ 3 ϭ
4.
108 2 LINEAR AND QUADRATIC FUNCTIONS
x ϭ 3. In Example 2, the algebraic solution again produced the exact solution,
x ϭ 111/34, while the graphical solution produced x ϭ 3.2647059, a seven-
decimal-place approximation to the solution. Some like to argue that this makes
the algebraic method superior to the graphical method. But exact solutions have
little relevance to most applications of mathematics and decimal approximations
are usually quite satisfactory.
We encourage you to choose the method that seems best to you, and when pos-
sible, use the other method to conﬁrm your answer. In a simple problem, like
Example 1, choose either method. In Example 2, we would recommend the alge-
braic method over the graphical method because of the complexity of the graphs.
We have not yet studied graphs of functions involving fractions with x in the
denominator. It was a fortunate accident that the intersection point was visible in
a standard viewing window.
We frequently encounter equations involving more than one variable. For
example, if L and W are the length and width of a rectangle, respectively, the area
of the rectangle is given by (see Fig. 3):
A ϭ LW
Depending on the situation, we may want to solve this equation for L or W. To
solve for W, we simply consider A and L to be constants and W to be a variable.
Then the equation A ϭ LW becomes a linear equation in W, which can be solved
easily by dividing both sides by L:
Solving an Equation with More than One Variable
Solve for P in terms of the other variables: A ϭ P ϩ Prt
S o l u t i o n Think of A, r, and t as constants.
Factor to isolate P.
Divide both sides by 1 ϩ rt.
Restriction: 1 ϩ rt 0
Solve for r in terms of the other variables: A ϭ P ϩ Prt
Solving Linear Inequalities
Now we want to turn our attention to inequalities. Any inequality that can be reduced
to one of the four forms in (1) is called a linear inequality in one variable.
M A T C H E D P R O B L E M
3
P ϭ
A
1 ϩ rt
A
1 ϩ rt
ϭ P
A ϭ P(1 ϩ rt)
A ϭ P ϩ Prt
E X A M P L E
3
W ϭ
A
L
L 0
A ϭ LW W
L
FIGURE 3
Area of a rectangle.
5.
Explore/Discuss
2
2-2 Linear Equations and Inequalities 109
mx ϩ b Ͼ 0
mx ϩ b Ն 0
mx ϩ b Ͻ 0
mx ϩ b Յ 0
As was the case with equations, the solution set of an inequality is the set of
all values of the variable that make the inequality a true statement. Each element
of the solution set is called a solution. Two inequalities are said to be equivalent
if they have the same solution set.
Associated with the linear equation and inequalities
3x Ϫ 12 ϭ 0 3x Ϫ 12 Ͻ 0 3x Ϫ 12 Ͼ 0
is the linear function
f(x) ϭ 3x Ϫ 12
(A) Graph the function f.
(B) From the graph of f determine the values of x for which
f(x) ϭ 0 f(x) Ͻ 0 f(x) Ͼ 0
(C) How are the answers to part B related to the solutions of
3x Ϫ 12 ϭ 0 3x Ϫ 12 Ͻ 0 3x Ϫ 12 Ͼ 0
As you discovered in Explore/Discuss 2, solving inequalities graphically is both
intuitive and efﬁcient. On the other hand, algebraic methods can become quite com-
plicated. So we will emphasize the graphical approach when solving inequalities.
Solving a Linear Inequality
Solve and graph on a number line: 0.5x ϩ 1 Յ 0
S o l u t i o n The graph of f(x) ϭ 0.5x ϩ 1 is shown in Figure 4. It is clear from the graph
that f(x) is negative to the left of x ϭ Ϫ2 and positive to the right. Thus, the solu-
tion set of the inequality
0.5x ϩ 1 Յ 0
is
x Յ Ϫ2 or (Ϫϱ, Ϫ2]
E X A M P L E
4
Linear inequalities (1)
Ϫ10
Ϫ10
10
10
FIGURE 4
f(x) ϭ 0.5x ϩ 1.
6.
Explore/Discuss
3
110 2 LINEAR AND QUADRATIC FUNCTIONS
Figure 5 shows a graph of the solution set on a number line. A similar graph can
be produced on most graphing utilities by entering y1 ϭ 0.5x ϩ 1 Յ 0 (Fig. 6).
The expression 0.5x ϩ 1 Յ 0 is assigned the value 1 for those values of x that
make it a true statement and the value 0 for those values of x that make it a false
statement.
Solve and graph on a number line: 2x Ϫ 6 Ն 0
Associated with the following equations and inequalities
ϩ 4 ϭ x Ϫ 2
ϩ 4 Ͻ x Ϫ 2
ϩ 4 Ͼ x Ϫ 2
are the two linear functions
f(x) ϭ ϩ 4 and g(x) ϭ x Ϫ 2
(A) Graph both f and g in the same viewing window.
(B) From the graph in part A determine the value(s) of x for which
f(x) ϭ g(x) f(x) Ͻ g(x) f(x) Ͼ g(x)
(C) How are the answers to part B related to the solutions of
ϩ 4 ϭ x Ϫ 2
ϩ 4 Ͻ x Ϫ 2
ϩ 4 Ͼ x Ϫ 2
Most inequalities can be solved graphically. If you need to algebraically
manipulate an inequality, Theorem 1 lists the properties that govern operations on
inequalities.
Ϫ1
2x
Ϫ1
2x
Ϫ1
2x
Ϫ1
2x
Ϫ1
2x
Ϫ1
2x
Ϫ1
2x
M A T C H E D P R O B L E M
4
FIGURE 6FIGURE 5
Ϫ10
Ϫ10
10
10
0Ϫ5 5Ϫ2
x]
7.
T H E O R E M
1
2-2 Linear Equations and Inequalities 111
INEQUALITY PROPERTIES
An equivalent inequality will result and the sense will remain the same,
if each side of the original inequality
1. Has the same real number added to or subtracted from it.
2. Is multiplied or divided by the same positive number.
An equivalent inequality will result and the sense will reverse, if each
side of the original inequality
3. Is multiplied or divided by the same negative number.
Note: Multiplication by 0 and division by 0 are not permitted.
To gain some experience with these properties, we will solve the next exam-
ple two ways, algebraically and graphically.
Solving a Double Inequality
Solve and graph on a number line: Ϫ3 Յ 4 Ϫ 7x Ͻ 18
S o l u t i o n To solve algebraically, we perform operations on the double inequality until we
have isolated x in the middle with a coefﬁcient of 1.
Subtract 4 from each member.
or or (2)
To solve graphically, enter y1 ϭ Ϫ3, y2 ϭ 4 Ϫ 7x, y3 ϭ 18, graph [Fig. 7(a)], and
ﬁnd the intersection points [Figs. 7(b) and 7(c)]. It is clear from the graph that y2
is between y1 and y3 for x between Ϫ2 and 1. Since y2 ϭ Ϫ3 at x ϭ 1, we include
1 in the solution set, obtaining the same solution, as shown in equation (2).
(a) (b) (c)
Ϫ15
Ϫ5
30
5
Ϫ15
Ϫ5
30
5
Ϫ15
Ϫ5
30
5
y3 ϭ 18
y1 ϭ Ϫ3
y2 ϭ 4 Ϫ 7xFIGURE 7
xx
0Ϫ5 1 5Ϫ2
( ]
(Ϫ2, 1]Ϫ2 Ͻ x Յ 11 Ն x Ͼ Ϫ2
Divide each member by Ϫ7 and
reverse each inequality.
Ϫ7
؊7
Ն
Ϫ7x
؊7
Ͼ
14
؊7
Ϫ7 Յ Ϫ7x Ͻ 14
Ϫ3 ؊ 4 Յ 4 Ϫ 7x ؊ 4 Ͻ 18 ؊ 4
Ϫ3 Յ 4 Ϫ 7x Ͻ 18
E X A M P L E
5
8.
Explore/Discuss
112 2 LINEAR AND QUADRATIC FUNCTIONS
Solve algebraically and graphically and graph on a number line:
Ϫ3 Ͻ 7 Ϫ 2x Յ 7
Solving Equations and Inequalities Involving
Absolute Value
Recall the deﬁnition of the absolute value function (see Section 1-4):
f(x) ϭ ͉x͉ ϭ
(A) Graph the absolute value function f(x) ϭ ͉x͉ and the constant func-
tion g(x) ϭ 3 in the same viewing window.
(B) From the graph in part A, determine the values of x for which
͉x͉ Ͻ 3 ͉x͉ ϭ 3 ͉x͉ Ͼ 3
(C) Discuss methods for using the deﬁnition of ͉x͉ and algebraic tech-
niques to solve part B.
The algebraic solution of an equation or inequality involving the absolute
value function usually must be broken down into two or more cases. For exam-
ple, to solve the equation
͉x Ϫ 4͉ ϭ 2 (3)
we consider two cases:
x Ϫ 4 ϭ 2 or x Ϫ 4 ϭ Ϫ2
x ϭ 6 x ϭ 2
We can also solve equation (3) graphically by graphing y1 ϭ ͉x Ϫ 4͉ and y2 ϭ 2,
and ﬁnding their intersection points, as shown in Figure 8.
(a) (b)
Algebraic solutions for inequalities involving absolute values can become
quite involved. However, as the next example illustrates, even problems that
appear to be complicated are easily solved with a graphing utility.
Ϫ10
Ϫ10
10
10
Ϫ10
Ϫ10
10
10
y2 ϭ 2
y1 ϭ ͉x Ϫ 4͉FIGURE 8
Graphical solution of ͉x Ϫ 4͉ ϭ 2.
Ά
Ϫx
x
if x Ͻ 0
if x Ն 0
M A T C H E D P R O B L E M
5
4
9.
2-2 Linear Equations and Inequalities 113
Solving Absolute Value Problems Graphically
Solve graphically. Write solutions in both inequality and interval notation and
graph on a number line.
(A) ͉2x Ϫ 5͉ Ͼ 4 (B) ͉0.5x ϩ 2͉ Ն 3x Ϫ 5
S o l u t i o n s (A) Graph y1 ϭ ͉2x Ϫ 5͉ and y2 ϭ 4 in the same viewing window and ﬁnd
the intersection points (Fig. 9). Examining the graphs in Figure 9, we see if
x Ͻ 0.5 or x Ͼ 4.5, then the graph of y1 is above the graph of y2. Thus, the
solution is
x Ͻ 0.5 or x Ͼ 4.5 Inequality notation
(Ϫϱ, 0.5) ʜ (4.5, ϱ)* Interval notation
(a) (b)
(B) Figure 10 shows the appropriate graphs for the inequality ͉0.5x ϩ 2͉ Ն
3x Ϫ 5. The graph in Figure 10 shows that y1 Ͼ y2 for x Ͻ 2.8. Since y1 ϭ
y2 for x ϭ 2.8, we must include this value of x in the solution set:
x Յ 2.8 or (Ϫϱ, 2.8]
Solve graphically and write solutions in both inequality and interval notation.
(A) ͉ ͉ Ն 2 (B) ͉2x Ϫ 5͉ Ͻ 0.4x ϩ 2
Application
Break-Even, Profit, and Loss
A recording company produces compact discs (CDs). One-time ﬁxed costs for
a particular CD are $24,000, which include costs such as recording, album
E X A M P L E
7
2
3x ϩ 1
M A T C H E D P R O B L E M
6
x
0Ϫ5 2.8 5
]
Ϫ10
Ϫ10
10
10
y2 ϭ 3x Ϫ 5
y1 ϭ ͉0.5x ϩ 2͉
Ϫ10
Ϫ10
10
10
Ϫ10
Ϫ10
10
10
y1 ϭ ͉2x Ϫ 5͉
y2 ϭ 4
FIGURE 10FIGURE 9
Ϫ5 0 5
4.50.5
() x
E X A M P L E
6
*The symbol ʜ denotes the union operation for sets. See Section A-8 for a discussion of interval notation
and set operations.
10.
114 2 LINEAR AND QUADRATIC FUNCTIONS
design, and promotion. Variable costs amount to $5.50 per CD and include
the manufacturing, distribution, and royalty costs for each disc actually
manufactured and sold to a retailer. The CD is sold to retail outlets for $8.00
each.
(A) Find the level of sales for which the company will break even. Describe
verbally and graphically the sales levels that result in a proﬁt and those
that result in a loss.
(B) Find the sales level that will produce a proﬁt of $20,000.
S o l u t i o n s (A) Let
x ϭ Number of CDs sold
C ϭ Total cost for producing x CDs
R ϭ Revenue (return) on sales of x CDs
Now form the cost and revenue functions.
C(x) ϭ Fixed costs ϩ Variable costs
ϭ 24,000 ϩ 5.5x Cost function
R(x) ϭ 8x Revenue function
The company will break even when revenue ϭ cost; that is, when R(x) ϭ
C(x). The solution to this equation is often referred to as the break-even
point. Graphs of both functions and their intersection point are shown in Fig-
ure 11. Examining this graph, we see that the company will break even if
they sell 9,600 CDs. If they sell more than 9,600 CDs, then revenue is greater
than cost, and the company will make a proﬁt. If they sell fewer than 9,600
CDs, then cost is greater than revenue and the company will lose money.
These sales levels are illustrated in Figure 12.
(B) The proﬁt function for this manufacturer is
P(x) ϭ R(x) Ϫ C(x)
ϭ 8x Ϫ (24,000 ϩ 5.5x)
ϭ 2.5x Ϫ 24,000
FIGURE 12FIGURE 11
x
y
20,000
200,000
y ϭ R(x) ϭ 8x
y ϭ C(x) ϭ 24,000 ϩ 5.5x
Loss Profit
Break-even point
9,600
Ϫ50,000
0
200,000
20,000
y2 ϭ 8x
y1 ϭ 24,000 ϩ 5.5x
11.
Explore/Discuss
5
2-2 Linear Equations and Inequalities 115
The sales level x that will produce a proﬁt of $20,000 is the solution of the
equation P(x) ϭ 20,000. Figure 13 shows a graphical solution of this linear
equation. Thus, we see that the company will make a proﬁt of $20,000 when
they sell 17,600 CDs.
Repeat Example 7 if ﬁxed costs are $28,000, variable costs are $6.60 per CD,
and the CDs are sold for $9.80 each.
(A) Find the x intercept of the proﬁt function in Example 7 (see Fig. 13).
(B) Discuss the relationship between the x intercept of the proﬁt func-
tion and the sales levels for which the company incurs a loss, breaks
even, or makes a proﬁt.
(C) In general, compare the graphical solutions of the inequalities
f(x) Ͼ g(x) and f(x) Ϫ g(x) Ͼ 0
Answers to Matched Problems
1. Ϫ2 2. Ϸ Ϫ0.5128205 3. r ϭ 4. x Ն 3 or [3, ϱ) 5. 0 Յ x Ͻ 5 or [0, 5)
6. (A) x Յ Ϫ4.5 or x Ն 1.5; (Ϫϱ, Ϫ4.5] ʜ [1.5, ϱ)
(B) 1.25 Ͻ x Ͻ 4.375; (1.25, 4.375)
7. (A) The company breaks even if they sell 8,750 CDs,
makes a proﬁt if they sell more than 8,750 CDs,
and loses money if they sell less than 8,750 CDs.
(B) The company must sell 15,000 CDs to make a
proﬁt of $20,000.
xx
0 5Ϫ5
1.25 4.375
( )
0
1.5
5
Ϫ4.5
] [ xx
Ϫ5
xx
0Ϫ5 5
)[x
0Ϫ5 3 5
[ x
A Ϫ P
Pt
Ϫ
20
39
M A T C H E D P R O B L E M
7
Ϫ50,000
0
50,000
20,000
y3 ϭ 2.5x Ϫ 24,000
y4 ϭ 20,000FIGURE 13
x
y
20,0008,750
200,000
Loss Profit
Break-even point
y ϭ C(x) ϭ 28,000 ϩ 6.6x
y ϭ R(x) ϭ 9.8x
12.
116 2 LINEAR AND QUADRATIC FUNCTIONS
EXERCISE 2-2
A
Use the graphs of functions u and v in the ﬁgure to solve the
equations and inequalities in Problems 1–8. (Assume the
graphs continue as indicated beyond the portions shown here.)
Express solutions to inequalities in interval notation.
1. u(x) ϭ 0 2. v(x) ϭ 0
3. u(x) ϭ v(x) 4. u(x) Ϫ v(x) ϭ 0
5. u(x) Ͼ 0 6. v(x) Ն 0
7. v(x) Ն u(x) 8. v(x) Ͻ 0
Solve Problems 9–14 algebraically and check graphically.
9. 3(x ϩ 2) ϭ 5(x Ϫ 6) 10. 5x ϩ 10(x Ϫ 2) ϭ 40
11. 5 ϩ 4(t Ϫ 2) ϭ 2(t ϩ 7) ϩ 1
12. 5w Ϫ (7w Ϫ 4) Ϫ 2 ϭ 5 Ϫ (3w ϩ 2)
13. 5 Ϫ 14.
Solve Problems 15–20 algebraically and check graphically.
Represent each solution using inequality notation, interval
notation, and a graph on a real number line.
15. 7x Ϫ 8 Ͻ 4x ϩ 7 16. 4x ϩ 8 Ն x Ϫ 1
17. Ϫ5t Ͻ Ϫ10 18. Ϫ7n Ն 21
19. Ϫ4 Ͻ 5t ϩ 6 Յ 21 20. 2 Յ 3m Ϫ 7 Ͻ 14
B
In Problems 21–36, solve each equation or inequality. When
applicable, write answers using both inequality notation and
interval notation.
21. ͉y Ϫ 5͉ ϭ 3 22. ͉x ϩ 1͉ ϭ 5
23. ͉5t ϩ 3͉ Յ 7 24. ͉2w Ϫ 9͉ Ͻ 6
25. 26.
27. Ϫ12 Ͻ (2 Ϫ x) Յ 24 28. 24 Յ (x Ϫ 5) Ͻ 36
2
3
3
4
2
3x
ϩ
1
2
ϭ
4
x
ϩ
4
3
1
m
Ϫ
1
9
ϭ
4
9
Ϫ
2
3m
x ϩ 3
4
Ϫ
x Ϫ 4
2
ϭ
3
8
2x Ϫ 1
4
ϭ
x ϩ 2
3
x
y
y ϭ u(x)
y ϭ v(x)
e fdc
b
a
29. ͉3t Ϫ 7͉ ϭ 30. ͉2s ϩ 3͉ ϭ 6 Ϫ 0.5s
31. ͉1.5x ϩ 6͉ Ͼ 0.3x ϩ 7.5 32. ͉7 Ϫ 2x͉ Ն x Ϫ 0.8
33. 34.
35. 6 Ͻ ͉x Ϫ 2͉ ϩ ͉x ϩ 1͉ Ͻ 12
36. ͉x ϩ 1͉ Ϫ ͉x Ϫ 2͉ Ͻ 0.4x
In Problems 37–44, solve for the indicated variable in terms of
the other variables.
37. an ϭ a1 ϩ (n Ϫ 1)d for d (arithmetic progressions)
38. F ϭ ϩ 32 for C (temperature scale)
39. for f (simple lens formula)
40. for R1 (electric circuit)
41. A ϭ 2ab ϩ 2ac for a (surface area of a rectangular solid)
42. A ϭ 2ab ϩ 2ac ϩ 2bc for c
43. 44.
45. Discuss the relationship between the graphs of y1 ϭ x and
y2 ϭ
46. Discuss the relationship between the graphs of y1 ϭ ͉x͉ and
y2 ϭ
47. Discuss the possible signs of the numbers a and b given
that
(A) ab Ͼ 0 (B) ab Ͻ 0
(C) (D)
48. Discuss the possible signs of the numbers a, b, and c given
that
(A) abc Ͼ 0 (B)
(C) (D)
In Problems 49–52, replace each question mark with Ͻ or Ͼ
and explain why your choice makes the statement true.
49. If a Ϫ b ϭ 1, then a ? b.
50. If u Ϫ v ϭ Ϫ2, then u ? v.
51. If a Ͻ 0, b Ͻ 0, and Ͼ 1, then a ? b.
52. If a Ͼ 0, b Ͼ 0, and Ͼ 1, then a ? b.
b
a
b
a
a2
bc
Ͻ 0
a
bc
Ͼ 0
ab
c
Ͻ 0
a
b
Ͻ 0
a
b
Ͼ 0
͙x2
.
͙x2
.
x ϭ
3y ϩ 2
y Ϫ 3
for yy ϭ
2x Ϫ 3
3x ϩ 5
for x
1
R
ϭ
1
R1
ϩ
1
R2
1
f
ϭ
1
d1
ϩ
1
d2
9
5 C
2x
x ϩ 4
ϭ 7 Ϫ
6
x ϩ 4
2x
x Ϫ 3
ϭ 7 ϩ
4
x Ϫ 3
4
3
t ϩ
1
2
116 2 LINEAR AND QUADRATIC FUNCTIONS
13.
2-2 Linear Equations and Inequalities 117
C
Problems 53–56 are calculus-related. Solve and graph. Write
each solution using interval notation.
53. 0 Ͻ ͉x Ϫ 3͉ Ͻ 0.1 54. 0 Ͻ ͉x Ϫ 5͉ Ͻ 0.01
55. 0 Ͻ ͉x Ϫ c͉ Ͻ d 56. 0 Ͻ ͉x Ϫ 4͉ Ͻ d
57. What are the possible values of ?
58. What are the possible values of ?
APPLICATIONS
59. Sales Commissions. One employee of a computer store is
paid a base salary of $2,150 a month plus an 8% commis-
sion on all sales over $7,000 during the month. How much
must the employee sell in 1 month to earn a total of
$3,170 for the month?
60. Sales Commissions. A second employee of the computer
store in Problem 59 is paid a base salary of $1,175 a month
plus a 5% commission on all sales during the month.
(A) How much must this employee sell in 1 month to earn
a total of $3,170 for the month?
(B) Determine the sales level where both employees
receive the same monthly income. If employees can
select either of these payment methods, how would
you advise an employee to make this selection?
61. Approximation. The area A of a region is approximately
equal to 12.436. The error in this approximation is less
than 0.001. Describe the possible values of this area both
with an absolute value inequality and with interval
notation.
62. Approximation. The volume V of a solid is approxi-
mately equal to 6.94. The error in this approximation is
less than 0.02. Describe the possible values of this volume
both with an absolute value inequality and with interval
notation.
63. Break-Even Analysis. An electronics ﬁrm is planning to
market a new graphing calculator. The ﬁxed costs are
$650,000 and the variable costs are $47 per calculator.
The wholesale price of the calculator will be $63. For the
company to make a proﬁt, it is clear that revenues must be
greater than costs.
(A) How many calculators must be sold for the company
to make a proﬁt?
(B) How many calculators must be sold for the company
to break even?
(C) Discuss the relationship between the results in parts A
and B.
Խx Ϫ 1Խ
x Ϫ 1
x
ԽxԽ
64. Break-Even Analysis. A video game manufacturer is
planning to market a 64-bit version of its game machine.
The ﬁxed costs are $550,000 and the variable costs are
$120 per machine. The wholesale price of the machine
will be $140.
(A) How many game machines must be sold for the
company to make a proﬁt?
(B) How many game machines must be sold for the
company to break even?
(C) Discuss the relationship between the results in parts A
and B.
65. Break-Even Analysis. The electronics ﬁrm in Problem 63
ﬁnds that rising prices for parts increases the variable
costs to $50.5 per calculator.
(A) Discuss possible strategies the company might use to
deal with this increase in costs.
(B) If the company continues to sell the calculators for
$63, how many must they sell now to make a proﬁt?
(C) If the company wants to start making a proﬁt at the
same production level as before the cost increase,
how much should they increase the wholesale price?
66. Break-Even Analysis. The video game manufacturer in
Problem 64 ﬁnds that unexpected programming problems
increases the ﬁxed costs to $660,000.
(A) Discuss possible strategies the company might use to
deal with this increase in costs.
(B) If the company continues to sell the game machines
for $140, how many must they sell now to make a
proﬁt?
(C) If the company wants to start making a proﬁt at the
same production level as before the cost increase,
how much should they increase the wholesale price?
55 67. Signiﬁcant Digits. If N ϭ 2.37 represents a measurement,
then we assume an accuracy of 2.37 Ϯ 0.005. Express the
accuracy assumption using an absolute value inequality.
5 68. Signiﬁcant Digits. If N ϭ 3.65 ϫ 10Ϫ3
is a number from a
measurement, then we assume an accuracy of 3.65 ϫ 10Ϫ3
Ϯ 5 ϫ 10Ϫ6
. Express the accuracy assumption using an
absolute value inequality.
5 69. Finance. If an individual aged 65–69 continues to work
after Social Security beneﬁts start, beneﬁts will be reduced
when earnings exceed an earnings limitation. In 1989,
beneﬁts were reduced by $1 for every $2 earned in excess
of $8,880. Find the range of beneﬁt reductions for individ-
uals earning between $13,000 and $16,000.
5 70. Finance. Refer to Problem 69. In 1990 the law was
changed so that beneﬁts were reduced by $1 for every $3
earned in excess of $8,880. Find the range of beneﬁt re-
ductions for individuals earning between $13,000 and
$16,000.
14.
118 2 LINEAR AND QUADRATIC FUNCTIONS
71. Celsius/Fahrenheit. A formula for converting Celsius de-
grees to Fahrenheit degrees is given by the linear function
F ϭ ϩ 32
Determine to the nearest degree the Celsius range in tem-
perature that corresponds to the Fahrenheit range of 60°F
to 80°F.
72. Celsius/Fahrenheit. A formula for converting Fahrenheit
degrees to Celsius degrees is given by the linear function
C ϭ (F Ϫ 32)
Determine to the nearest degree the Fahrenheit range in
temperature that corresponds to a Celsius range of 20°C to
30°C.
5 73. Earth Science. In 1984, the Soviets led the world in
drilling the deepest hole in the Earth’s crust—more than
12 kilometers deep. They found that below 3 kilometers
the temperature T increased 2.5°C for each additional 100
meters of depth.
(A) If the temperature at 3 kilometers is 30°C and x is the
depth of the hole in kilometers, write an equation
using x that will give the temperature T in the hole at
any depth beyond 3 kilometers.
(B) What would the temperature be at 15 kilometers?
[The temperature limit for their drilling equipment
was about 300°C.]
5
9
9
5
C
(C) At what interval of depths will the temperature be
between 200°C and 300°C, inclusive?
5 74. Aeronautics. Because air is not as dense at high altitudes,
planes require a higher ground speed to become airborne.
A rule of thumb is 3% more ground speed per 1,000 feet of
elevation, assuming no wind and no change in air tempera-
ture. (Compute numerical answers to 3 signiﬁcant digits.)
(A) Let
Vs ϭ Takeoff ground speed at sea level for a particular
plane (in miles per hour)
A ϭ Altitude above sea level (in thousands of feet)
V ϭ Takeoff ground speed at altitude A for the same
plane (in miles per hour)
Write a formula relating these three quantities.
(B) What takeoff ground speed would be required at Lake
Tahoe airport (6,400 feet), if takeoff ground speed at
San Francisco airport (sea level) is 120 miles per
hour?
(C) If a landing strip at a Colorado Rockies hunting lodge
(8,500 feet) requires a takeoff ground speed of 125
miles per hour, what would be the takeoff ground
speed in Los Angeles (sea level)?
(D) If the takeoff ground speed at sea level is 135 miles
per hour and the takeoff ground speed at a mountain
resort is 155 miles per hour, what is the altitude of the
mountain resort in thousands of feet?
Section 2-3 Quadratic Functions
Quadratic Functions
Completing the Square
Properties of Quadratic Functions and Their Graphs
Applications
Quadratic Functions
The graph of the square function, h(x) ϭ x2
, is shown in Figure 1. Notice that the
graph is symmetric with respect to the y axis and that (0, 0) is the lowest point
on the graph. Let’s explore the effect of applying a sequence of basic transfor-
mations to the graph of h. (A brief review of Section 1-5 might prove helpful at
this point.)
h(x)
5
Ϫ5 5
x
FIGURE 1
Square function h(x) ϭ x2
.
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