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A Precalculus Review
CHAPTER 0
0.1
Page 0-2
THE REAL NUMBER LINE AND ORDER
■
■
■
■
Represent, classify, and order real numbers.
Use inequalities to represent sets of real numbers.
Solve inequalities.
Use inequalities to model and solve real-life problems.
The Real Number Line
Negative direction
(x decreases)
Positive direction
(x increases)
x
−4 −3 −2 −1
FIGURE 0.1
0
1
2
3
4
The Real Number Line
5
4
−2.6
x
−3
−2
−1
0
1
2
3
Every point on the real number line
corresponds to one and only one real number.
7
−3
1.85
2.6 13
5
7
3
x
−3
−2
−1
0
1
2
3
Every real number corresponds to one and
only one point on the real number line.
e
π
x
−1
0
FIGURE 0.3
1
2
3
5
4
37
1.85 20
Such numbers are called rational. Rational numbers have either terminating or
infinitely repeating decimal representations.
Terminating Decimals
2
0.4
5
7
0.875
8
FIGURE 0.2
2
Real numbers can be represented with a coordinate system called the real number line (or x-axis), as shown in Figure 0.1. The positive direction (to the right)
is denoted by an arrowhead and indicates the direction of increasing values of x.
The real number corresponding to a particular point on the real number line is
called the coordinate of the point. As shown in Figure 0.1, it is customary to label
those points whose coordinates are integers.
The point on the real number line corresponding to zero is called the origin.
Numbers to the right of the origin are positive, and numbers to the left of the origin are negative. The term nonnegative describes a number that is either positive
or zero.
The importance of the real number line is that it provides you with a
conceptually perfect picture of the real numbers. That is, each point on the real
number line corresponds to one and only one real number, and each real number
corresponds to one and only one point on the real number line. This type of relationship is called a one-to-one correspondence and is illustrated in Figure 0.2.
Each of the four points in Figure 0.2 corresponds to a real number that can
be expressed as the ratio of two integers.
Infinitely Repeating Decimals
1
0.333 . . . 0.3*
3
12
1.714285714285 . . . 1.714285
7
Real numbers that are not rational are called irrational, and they cannot be
represented as the ratio of two integers (or as terminating or infinitely repeating
decimals). So, a decimal approximation is used to represent an irrational number.
Some irrational numbers occur so frequently in applications that mathematicians
have invented special symbols to represent them. For example, the symbols 2,
, and e represent irrational numbers whose decimal approximations are as
shown. (See Figure 0.3.)
2 1.4142135623
3.1415926535
e 2.7182818284
*The bar indicates which digit or digits repeat infinitely.
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SECTION 0.1
0-3
The Real Number Line and Order
Order and Intervals on the Real Number Line
One important property of the real numbers is that they are ordered: 0 is less
than 1, 3 is less than 2.5, is less than 22
7 , and so on. You can visualize this
property on the real number line by observing that a is less than b if and only if
a lies to the left of b on the real number line. Symbolically, “a is less than b” is
denoted by the inequality
a < b.
For example, the inequality 43 < 1 follows from the fact that 34 lies to the left of 1
on the real number line, as shown in Figure 0.4.
3
4
lies to the left of 1, so
3
4
−1
0
3
4
< 1.
1
x
1
2
FIGURE 0.4
When three real numbers a, x, and b are ordered such that a < x and x < b,
we say that x is between a and b and write
a < x < b.
x is between a and b.
The set of all real numbers between a and b is called the open interval between
a and b and is denoted by a, b. An interval of the form a, b does not contain
the “endpoints” a and b. Intervals that include their endpoints are called closed
and are denoted by a, b. Intervals of the form a, b and a, b are neither open
nor closed. Figure 0.5 shows the nine types of intervals on the real number line.
Intervals that are neither open nor closed
Open interval
(− ∞, a)
(a, b]
(a, b)
a
b
a
b
a<x≤b
a<x<b
Infinite intervals
a
(b, ∞)
a
b
x>b
(− ∞, a]
[b, ∞)
[a, b)
Closed interval
a
[a, b]
a
b
a
b
x≤a
(− ∞, ∞)
b
a≤x≤b
STUDY
Intervals on the Real Number Line
TIP
Note that a square bracket is used to denote “less than or equal to” ≤ or
“greater than or equal to” ≥. Furthermore, the symbols and denote
positive and negative infinity. These symbols do not denote real numbers;
they merely let you describe unbounded conditions more concisely. For
instance, the interval b, is unbounded to the right because it includes all
real numbers that are greater than or equal to b.
b
x≥b
a≤x<b
a
FIGURE 0.5
b
x<a
a
b
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A Precalculus Review
CHAPTER 0
Solving Inequalities
STUDY
TIP
Notice the differences between
Properties 3 and 4. For example,
3 < 4 ⇒ 32 < 42
and
In calculus, you are frequently required to “solve inequalities” involving variable
expressions such as 3x 4 < 5. The number a is a solution of an inequality if
the inequality is true when a is substituted for x. The set of all values of x that satisfy an equality is called the solution set of the inequality. The following properties are useful for solving inequalities. (Similar properties are obtained if < is
replaced by ≤ and > is replaced by ≥.)
3 < 4 ⇒ 32 > 42.
Properties of Inequalities
Let a, b, c, and d be real numbers.
1. Transitive property: a < b and b < c
ALGEBRA
a < c
ac < bd
2. Adding inequalities: a < b and c < d
REVIEW
Once you have solved an inequality, it is a good idea to check some
x-values in your solution set to see
whether they satisfy the original
inequality. You might also check
some values outside your solution
set to verify that they do not satisfy
the inequality. For example, Figure
0.6 shows that when x 0 or
x 2 the inequality is satisfied,
but when x 4 the inequality is
not satisfied.
3. Multiplying by a (positive) constant: a < b
ac < bc,
c > 0
4. Multiplying by a (negative) constant: a < b
ac > bc,
c < 0
5. Adding a constant: a < b
6. Subtracting a constant: a < b
ac < bc
ac < bc
Note that you reverse the inequality when you multiply by a negative number. For
example, if x < 3, then 4x > 12. This principle also applies to division by a
negative number. So, if 2x > 4, then x < 2.
EXAMPLE 1
Solving an Inequality
Find the solution set of the inequality 3x 4 < 5.
SOLUTION
For x = 0, 3(0) − 4 = − 4.
For x = 2, 3(2) − 4 = 2.
For x = 4, 3(4) − 4 = 8.
x
−1
0
1
2
Solution set for
3x − 4 < 5
FIGURE 0.6
3
4
5
6
7
8
3x 4
3x 4 4
3x
1
3x
3
x
< 5
Write original inequality.
< 54
Add 4 to each side.
< 9
Simplify.
1
< 9
3
< 3
1
Multiply each side by 3 .
Simplify.
So, the solution set is the interval , 3, as shown in Figure 0.6.
TRY
IT
1
Find the solution set of the inequality 2x 3 < 7.
In Example 1, all five inequalities listed as steps in the solution have the same
solution set, and they are called equivalent inequalities.
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SECTION 0.1
0-5
The Real Number Line and Order
The inequality in Example 1 involves a first-degree polynomial. To solve
inequalities involving polynomials of higher degree, you can use the fact that a
polynomial can change signs only at its real zeros (the real numbers that make the
polynomial zero). Between two consecutive real zeros, a polynomial must be
entirely positive or entirely negative. This means that when the real zeros of a
polynomial are put in order, they divide the real number line into test intervals
in which the polynomial has no sign changes. That is, if a polynomial has the factored form
x r1x r2, . . . , x rn ,
r1 < r2 < r3 < . . . < rn
then the test intervals are
, r1, r1, r2, . . . , rn1, rn , and rn, .
For example, the polynomial
x2 x 6 x 3x 2
can change signs only at x 2 and x 3. To determine the sign of the
polynomial in the intervals , 2, 2, 3, and 3, , you need to test only
one value from each interval.
EXAMPLE 2
Solving a Polynomial Inequality
Find the solution set of the inequality x2 < x 6.
SOLUTION
x2 < x 6
2
x x6 < 0
x 3x 2 < 0
2 < x < 3,
x
Sign
< 0?
3
No
Write original inequality.
2
0
No
Polynomial form
1
Yes
0
Yes
1
Yes
2
Yes
3
0
No
4
No
Factor.
So, the polynomial x2 x 6 has x 2 and x 3 as its zeros. You can
solve the inequality by testing the sign of the polynomial in each of the following
intervals.
x < 2,
Sign of x 3x 2
x > 3
To test an interval, choose a representative number in the interval and compute
the sign of each factor. For example, for any x < 2, both of the factors x 3
and x 2 are negative. Consequently, the product (of two negative numbers) is
positive, and the inequality is not satisfied in the interval
x < 2.
x
A convenient testing format is shown in Figure 0.7. Because the inequality is
satisfied only by the center test interval, you can conclude that the solution set is
given by the interval
2 < x < 3.
TRY
IT
Solution set
2
Find the solution set of the inequality x2 > 3x 10.
−2
No
(−)(−) > 0
FIGURE 0.7
3
Yes
No
(−)(+) < 0 (+)(+) > 0
Is x 3x 2 < 0?
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A Precalculus Review
Application
Inequalities are frequently used to describe conditions that occur in business and
science. For instance, the inequality
144 ≤ W ≤ 180
describes the recommended weight W for a man whose height is 5 feet 10
inches. Example 3 shows how an inequality can be used to describe the production level of a manufacturing plant.
EXAMPLE 3
Production Levels
In addition to fixed overhead costs of $500 per day, the cost of producing x units
of an item is $2.50 per unit. During the month of August, the total cost of
production varied from a high of $1325 to a low of $1200 per day. Find the high
and low production levels during the month.
SOLUTION Because it costs $2.50 to produce one unit, it costs 2.5x to produce
x units. Furthermore, because the fixed cost per day is $500, the total daily cost
of producing x units is
C 2.5x 500.
Now, because the cost ranged from $1200 to $1325, you can write the following.
1200
1200 500
700
700
2.5
280
2.5x 500
≤ 2.5x 500 500
≤
2.5x
2.5x
≤
2.5
≤
≤
x
≤ 1325
Write original inequality.
≤ 1325 500 Subtract 500 from each side.
≤ 825
825
≤
2.5
≤ 330
Simplify.
Divide each side by 2.5.
Simplify.
So, the daily production levels during the month of August varied from a low of
280 units to a high of 330 units, as shown in Figure 0.8.
Each day’s production
during the month
fell in this interval.
Low daily
production
280
High daily
production
330
x
0
100
200
300
400
500
FIGURE 0.8
TRY
IT
3
Use the information in Example 3 to find the high and low production levels
if, during October, the total cost of production varied from a high of $1500
to a low of $1000 per day.
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The Real Number Line and Order
SECTION 0.1
E X E R C I S E S
0 . 1
In Exercises 1–10, determine whether the real number is rational
or irrational.
2. 3678
*1. 0.7
3
3.
2
4. 32 1
5. 4.3451
6.
3 64
7. 8. 0.8177
3 60
9. 22
7
10. 2e
In Exercises 11–14, determine whether each given value of x
satisfies the inequality.
11. 5x 12 > 0
(a) x 3
(c) x 12. x 1 <
5
2
(b) x 3
3
(d) x 2
2x
3
(b) x 4
(c) x 4
(d) x 3
30. Physiology The maximum heart rate of a person in normal health is related to the person’s age by the equation
r 220 A
where r is the maximum heart rate in beats per minute and
A is the person’s age in years. Some physiologists recommend that during physical activity a person should strive to
increase his or her heart rate to at least 60% of the maximum heart rate for sedentary people and at most 90% of the
maximum heart rate for highly fit people. Express as an
interval the range of the target heart rate for a 20-year-old.
The revenue for selling x units of a product is
R 115.95x
and the cost of producing x units is
x2
< 2
4
C 95x 750.
(a) x 4
(b) x 10
(c) x 0
7
(d) x 2
14. 1 <
29. Biology: pH Values The pH scale measures the concentration of hydrogen ions in a solution. Strong acids produce low pH values, while strong bases produce high pH
values. Represent the following approximate pH values on
a real number line: hydrochloric acid, 0.0; lemon juice,
2.0; oven cleaner, 13.0; baking soda, 9.0; pure water, 7.0;
black coffee, 5.0. (Source: Adapted from Levine/Miller,
Biology: Discovering Life, Second Edition)
31. Profit
(a) x 0
13. 0 <
0-7
To obtain a profit, the revenue must be greater than the
cost. For what values of x will this product return a profit?
32. Sales A doughnut shop at a shopping mall sells a dozen
doughnuts for $3.50. Beyond the fixed cost (for rent, utilities, and insurance) of $170 per day, it costs $1.75 for
enough materials (flour, sugar, etc.) and labor to produce
each dozen doughnuts. If the daily profit varies between
$40 and $250, between what levels (in dozens) do the daily
sales vary?
3x
≤ 1
2
(a) x 0
(b) x 5
(c) x 1
(d) x 5
In Exercises 15–28, solve the inequality and sketch the graph of
the solution on the real number line.
33. Reimbursement A pharmaceutical company reimburses their sales representatives $0.35 per mile driven and
$100 for meals per week. The company allocates from
$200 to $250 per sales representative each week. What are
the minimum and maximum numbers of miles the company expects each representative to drive each week?
15. x 5 ≥ 7
16. 2x > 3
17. 4x 1 < 2x
18. 2x 7 < 3
19. 4 2x < 3x 1
20. x 4 ≤ 2x 1
21. 4 < 2x 3 < 4
22. 0 ≤ x 3 < 5
3
1
23. > x 1 >
4
4
x
24. 1 < < 1
3
34. Area A square region is to have an area of at least 500
square meters. What must the length of the sides of the
region be?
x
x
> 5
2 3
In Exercises 35 and 36, determine whether each statement is true
or false, given a < b.
25.
x
x
> 5
2 3
27. 2x 2 x < 6
26.
28. 2x2 1 < 9x 3
* The answers to the odd-numbered and selected even exercises are
given in the back of the text. Worked-out solutions to the oddnumbered exercises are given in the Student Solutions Guide.
35. (a) 2a < 2b
36. (a) a 4 < b 4
(b) a 2 < b 2
(b) 4 a < 4 b
(c) 6a < 6b
(c) 3b < 3a
(d)
1 1
<
a b
(d)
a b
<
4 4