Download Chapter 1 Section 6 - Canton Local Schools

Chapter 1
Equations and
Inequalities
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All rights reserved
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SECTION 1.6
Linear Inequalities
OBJECTIVES
1
2
3
4
Learn the vocabulary for discussing inequalities.
Solve and graph linear inequalities.
Solve and graph a compound inequality.
Solve and graph an inequality involving the
reciprocal of a linear expression.
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2
INEQUALITIES
Equation
Replace =
by
Inequality
x=5
<
x<5
3x + 2 = 14
≤
3x + 2 ≤ 14
5x + 7 = 3x + 23
>
5x + 7 > 3x + 23
x2 = 0
≥
x2 ≥ 0
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Definitions
An inequality is a statement that one algebraic
expression is less than, or is less than or equal
to, another algebraic expression.
The domain of a variable in an inequality is the
set of all real numbers for which both sides of
the inequality are defined.
The solutions of the inequality are the real
numbers that result in a true statement when
those numbers are substituted for the variable
in the inequality.
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Definitions
To solve an inequality means to find all solutions
of the inequality–that is, the solution set.
The solution sets are intervals, and we
frequently graph the solutions sets for
inequalities in one variable on a number line.
The graph of the inequality x < 5 is the interval
(–∞, 5) and is shown here.
)
5
x < 5, or (–∞, 5)
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Definitions
A conditional inequality such as x < 5 has in
its domain at least one solution and at least one
number that is not a solution.
An inconsistent inequality is one in which no
real number satisfies it.
An identity is an inequality that is satisfied by
every real number in the domain.
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NONNEGATIVE IDENTITY
x2  0
for any real number x.
Because x2 = x · x is the product of either (1)
two positive factors, (2) two negative factors,
or (3) two zero factors, x2 is always either a
positive number or zero. That is, x2 is never
negative, or is nonneagtive.
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EQUIVALENT INEQUALITIES
Two inequalities that have exactly the same
solution set are called equivalent inequalities.
The following operations produce equivalent
inequalities.
1. Simplifying one or both sides of an
inequality by combining like terms and
eliminating parentheses
2. Adding or subtracting the same expression on
both sides of the inequality
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If C represents a real number, then the
following inequalities are all equivalent.
Sign
of C
positive
Inequality
Sense
Example
A<B
<
3x < 12
A · C < B · C Unchanged
positive
A B

C C
Unchanged
negative
A·C>B·C
Reversed
negative
A B

C C
Reversed
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1
 3x   12 
3
3
3 x 12

3
3
1
1
  3x    12 
3
3
3x
12

3
3
9
Linear Inequalities
A linear inequality in one variable is an
inequality that is equivalent to one of the
forms
ax + b < 0 or ax + b ≤ 0,
where a and b represent real numbers and
a ≠ 0.
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EXAMPLE 1
Solving and Graphing Linear Inequalities
Solve each inequality and graph its solution set.
a. 7 x  11  2  x  3
b. 8  3x  2
Solution
a.
7 x  11  2  x  3
7 x  11  11  2  x  3  11
7x  2x  5
7x  2x  2x  5  2x
5x  5
5x 5

5 5
x 1
The solution set is {x|x < 1}, or (–∞, 1).
)
–1 0 1
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EXAMPLE 1
Solving and Graphing Linear Inequalities
Solution continued
b. 8  3 x  2
8  3x  8  2  8
3 x  6
3 x 6

3 3
x2
The solution set is {x|x ≥ 2}, or [2, ∞).
[
0 1 2
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EXAMPLE 2
Calculating the Results of the Bermuda
Triangle Experiment
In the introduction to this section, we discussed
an experiment to test the reliability of compass
settings and flight by automatic pilot along one
edge of the Bermuda Triangle. The plane is 150
miles along its path from Miami to Bermuda,
cruising at 300 miles per hour, when it notifies
the tower that it is now set on automatic pilot.
The entire trip is 1035 miles, and we want
to determine how much time we should let pass
before we become concerned that the plane has
encountered trouble.
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EXAMPLE 2
Calculating the Results of the Bermuda
Triangle Experiment
Solution
Let t = time elapsed since plane on autopilot
300t = distance plane flown in t hours
150 + 300t = plane’s distance from Miami after
t hours
Plane’s distance
from Miami
≥
Distance from
Miami to Bermuda
150  300t  1035
150  300t  150  1035  150
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EXAMPLE 2
Calculating the Results of the Bermuda
Triangle Experiment
Solution continued
300t  885
300t 885

300 300
t  2.95
Since 2.95 is roughly 3 hours, the tower will
suspect trouble if the plane has not arrived in
3 hours.
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EXAMPLE 3
Solving an Inequality
Write the solution set of each inequality.
Solution
The last inequality is always true. So the solution
set of the original inequality is (–∞, ∞).
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EXAMPLE 3
Solving an Inequality
Solution continued
The resulting inequality is always false. So the
solution set of the original inequality is .
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EXAMPLE 4
Solving & Graphing a Compound Inequality
Graph and write the solution set of the compound
inequality.
Solution
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EXAMPLE 4
Solving & Graphing a Compound Inequality
Solution continued
Graph each inequality and select the union of the
two intervals.
x  –3
x>2
x  –3 or x > 2
The solution set of the compound inequality is
(–∞, –3]  (2, ∞).
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EXAMPLE 5
Solving & Graphing a Compound Inequality
Graph and write the solution set of the compound
inequality.
Solution
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EXAMPLE 5
Solving & Graphing a Compound Inequality
Solution continued
Graph each inequality and select the interval
common to both inequalities.
x<5
x ≥ –2
x < 5 and x ≥ –2
The solution set of the compound inequality is
the interval [–2, 5).
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EXAMPLE 6
Solving & Graphing a Compound Inequality
Solve the inequality 5 x  2 x  3  9 and
graph its solution set.
Solution
First, solve the inequalities separately.
5  2 x  3
5  3  2 x  3  3
8 2 x

2
2
4  x
2x  3  9
2x  3  3  9  3
2x 6

2 2
x3
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EXAMPLE 6
Solving & Graphing a Compound Inequality
Solution continued
The solution to the original inequalities consists
of all real numbers x such that – 4 < x and x ≤ 3.
We can write this as {x| – 4 < x ≤ 3}. In
interval notation we write (– 4, 3].
(
]
– 5 – 4 –3 –2 –1 0 1 2 3 4
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EXAMPLE 6
Solving & Graphing a Compound Inequality
Solution continued
We can accomplish this solution more efficiently
by working on both inequalities at the same time
as follows.
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RECIPROCAL SIGN PROPERTY
1
If x ≠ 0, x and
are either both positive or
x
1
negative. In symbols, if x > 0, then  0
x
1
and if x < 0, then  0.
x
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EXAMPLE 7
Solving and Graphing an Inequality by
Using the Reciprocal Sign Property
Solve and graph
Solution
 3x  12 
1
1
0
3x  12
3x  12  0
 3x  12 
0
1
 0.
3x  12
3x 12

3
3
x4
The solution set is {x| x >4}, or in interval
notation (4, ∞).
(
0 1 2 3 4 5
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EXAMPLE 8
Finding the Interval of Values for a Linear
Expression
If –2 < x < 5, find real numbers a and b so that
a < 3x – 1 < b.
Solution
2  x  5
3  2   3 x  3  5 
6  3 x  15
6  1  3 x  1  15  1
7  3 x  1  14
We have a = –7 and b = 14.
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EXAMPLE 9
Finding a Fahrenheit Temperature from a
Celsius Range
The weather in London is predicted to range
between 10º and 20º Celsius during the threeweek period you will be working there. To
decide what kind of clothes to bring, you want to
convert the temperature range to Fahrenheit
temperatures. The formula for converting Celsius
temperature C to Fahrenheit temperature F is
9
F  C  32. What range of Fahrenheit
5
temperatures might you find in London during
your stay there?
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EXAMPLE 9
Finding a Fahrenheit Temperature from a
Celsius Range
Solution
Let C = temperature in Celsius degrees.
For the three weeks under consideration
10 ≤ C ≤ 20.
9
9
9
  10   C     20 
5
5
5
9
18  C  36
5
9
18  32  C  32  36  32
5
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EXAMPLE 9
Finding a Fahrenheit Temperature from a
Celsius Range
Solution continued
9
50  C  32  68
5
50  F  68
So, the temperature range from 10º to 20º
Celsius corresponds to a range from 50º to 68º
Fahrenheit.
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