Transcript Sec 3.1 - 1 - Palm Beach State College

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Sec 3.1 - 1
Chapter 3
Linear Inequalities and Absolute Value
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Sec 3.1 - 2
3.1
Linear Inequalities in One Variable
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Sec 3.1 - 3
3.1 Linear Inequalities in One Variable
Objectives
1.
Graph intervals on a number line.
2.
Solve linear inequalities using the addition property.
3.
Solve linear inequalities using the multiplication
property.
4.
Solve linear inequalities with three parts.
5.
Solve applied problems using linear inequalities.
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Sec 3.1 - 4
3.1 Linear Inequalities in One Variable
Graphing intervals on a number line
Solving inequalities is closely related to solving
equations. Inequalities are algebraic expressions
related by
We solve an inequality by finding all real numbers
solutions for it.
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Sec 3.1 - 5
3.1 Linear Inequalities in One Variable
Graphing intervals on a number line
One way to describe the solution set of an inequality
is by graphing.
We graph all the numbers satisfying x < –1 by placing
a right parenthesis at –1 on the number line and
drawing an arrow extending from the parenthesis to
the left. This arrow represents the fact that all
numbers less than –1 are part of the graph.
–5
–4
–3
–2
–1
0
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Sec 3.1 - 6
3.1 Linear Inequalities in One Variable
Interval Notation and the Infinity Symbol
The set of numbers less than –1 is an example of an
interval. We can write the solution set of this
inequality using interval notation.
• The
symbol does not actually represent a
number.
• A parenthesis is always used next to the infinity
symbol.
• The set of real numbers is written as
interval notation.
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Sec 3.1 - 7
3.1 Linear Inequalities in One Variable
EXAMPLE 1
Graphing Intervals Written In Interval
Notation on Number Lines
Write the inequality in interval notation and graph it.
This statement says that x can be any number greater than or equal
to 1. This interval is written
.
We show this on the number line by using a left bracket at 1 and
drawing an arrow to the right. The bracket indicates that the
number 1 is included in the interval.
–5
–4
–3
–2
–1
0
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Sec 3.1 - 8
3.1 Linear Inequalities in One Variable
EXAMPLE 2
Graphing Intervals Written In Interval
Notation on Number Lines
Write the inequality in interval notation and graph it.
This statement says that x can be any number greater than –2 and
less than or equal to 3. This interval is written
.
We show this on the number line by using a left parenthesis at –
2 and a right bracket at 3 and drawing a line between. The
parenthesis indicates that the number –2 is not included in the
interval and the bracket indicates that the 3 is included in the
interval.
–5
–4
–3
–2
–1
0
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Sec 3.1 - 9
3.1 Linear Inequalities in One Variable
Types of Intervals Summarized
Open Intervals
Set
Interval
Notation
Graph
a
a
b
b
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Sec 3.1 - 10
3.1 Linear Inequalities in One Variable
Types of Intervals Summarized
Half Open Intervals
Set
Interval
Notation
Graph
a
a
b
a
b
b
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Sec 3.1 - 11
3.1 Linear Inequalities in One Variable
Types of Intervals Summarized
Closed Interval
Set
Interval
Notation
Graph
a
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b
Sec 3.1 - 12
3.1 Linear Inequalities in One Variable
Linear Inequality
An inequality says that two expressions are not equal.
Linear Inequality
Examples:
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Sec 3.1 - 13
3.1 Linear Inequalities in One Variable
Solving Linear Inequalities Using the Addition Property
• Solving an inequality means to find all the numbers
that make the inequality true.
• Usually an inequality has a infinite number of
solutions.
• Solutions are found by producing a series of
simpler equivalent equations, each having the same
solution set.
• We use the addition and multiplication properties of
inequality to produce equivalent inequalities.
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Sec 3.1 - 14
3.1 Linear Inequalities in One Variable
Addition Property of Inequality
Addition Property of Inequality
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Sec 3.1 - 15
3.1 Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Check: Substitute –4 for x in the equation x – 5 = 9.
The result should be a true statement.
This shows –4
is a boundary
point.
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Sec 3.1 - 16
3.1 Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Now we have to test a number on each side of –4 to
verify that numbers greater than –4 make the inequality
true. We choose –3 and –5.
–5
–4
–3
–2
–1
0
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Sec 3.1 - 17
3.1 Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Check: Substitute 3 for m in the equation 3 + 7m = 8m.
The result should be a true statement.
This shows 3 is
a boundary
point.
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Sec 3.1 - 18
3.1 Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Now we have to test a number on each side of 3 to verify
that numbers less than or equal to 3 make the inequality
true. We choose 2 and 4.
–5
–4
–3
–2
–1
0
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Sec 3.1 - 19
3.1 Linear Inequalities in One Variable
Multiplication Property of Inequality
Multiplication Property of Inequality
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Sec 3.1 - 20
3.1 Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
Check: Substitute –8 for m in the equation 3m = –24.
The result should be a true statement.
This shows –8
is a boundary
point.
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Sec 3.1 - 21
3.1 Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
Now we have to test a number on each side of –8 to
verify that numbers greater than or equal to –8 make the
inequality true. We choose –9 and –7.
–16 –14 –12 –10
–8
–6
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–4
–2
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Sec 3.1 - 22
3.1 Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
Check: Substitute – 5 for k in the equation –7k = 35.
The result should be a true statement.
This shows –5
is a boundary
point.
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Sec 3.1 - 23
3.1 Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
Now we have to test a number on each side of –5 to
verify that numbers less than or equal to –5 make the
inequality true. We choose –6 and –4.
–16 –14 –12 –10
–8
–6
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–4
–2
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Sec 3.1 - 24
3.1 Linear Inequalities in One Variable
Solving a Linear Inequality
Steps used in solving a linear inequality are:
Step 1
Simplify each side separately. Clear
parentheses, fractions, and decimals using the
distributive property as needed, and combine
like terms.
Step 2
Isolate the variable terms on one side. Use
the additive property of inequality to get all
terms with variables on one side of the
inequality and all numbers on the other side.
Step 3
Isolate the variable. Use the multiplication
property of inequality to change the inequality to
the form x < k or x > k.
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Sec 3.1 - 25
3.1 Linear Inequalities in One Variable
Solving a Linear Inequality
Solve and graph the solution:
Step 1
Step 2
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Sec 3.1 - 26
3.1 Linear Inequalities in One Variable
Solving a Linear Inequality
Solve and graph the solution:
Step 3
–10
–9
–8
–7
–6
–5
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–4
–3
–2
–1
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3.1 Linear Inequalities in One Variable
Solving a Linear Inequality with Fractions
Solve and graph the solution:
First Clear
Fractions:
Multiply each side by the least common
denominator, 15.
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Sec 3.1 - 28
3.1 Linear Inequalities in One Variable
Solving a Linear Inequality with Fractions
Solve and graph the solution:
Step 1
Step 2
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Sec 3.1 - 29
3.1 Linear Inequalities in One Variable
Solving a Linear Inequality with Fractions
Solve and graph the solution:
Step 3
–16 –14 –12 –10
–8
–6
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–4
–2
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Sec 3.1 - 30
3.1 Linear Inequalities in One Variable
Solving Linear Inequalities with Three Parts
In some applications, linear inequalities have three parts.
When linear inequalities have three parts, it is important
to write the inequalities so that:
1. The inequality symbols point in the same
direction.
2. Both inequality symbols point toward the lesser
numbers.
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Sec 3.1 - 31
3.1 Linear Inequalities in One Variable
Solving a Three-Part Inequality
Solve and graph the solution:
This statement says that x – 2 is greater than or equal to
3 and less than or equal to 7.
To solve this inequality, we need to isolate the variable
x. To do this, we must add 2 to the expression, x – 2.
To produce an equivalent statement, we must also add
2 to the other two parts of the inequality as well.
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Sec 3.1 - 32
3.1 Linear Inequalities in One Variable
Solving a Three-Part Inequality
Solve and graph the solution:
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3.1 Linear Inequalities in One Variable
Solving a Three-Part Inequality
Solve and graph the solution:
–2
–1
0
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3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
In addition to the familiar phrases “less than” and
“greater than”, it is important to accurately interpret
the meaning of the following:
Word Expression
Interpretation
a is at least b
a is no less than b
a is at most b
a is no more than b
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Sec 3.1 - 35
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
A rectangle must have an area of at least 15 cm2 and no
more than 60 cm2. If the width of the rectangle is 3 cm,
what is the range of values for the length?
Step 1
Read the problem.
Step 2
Assign a variable. Let L = the length of the
rectangle.
Step 3
Write an inequality. Area equals width times
length, so area is 3L; and this amount must
be at least 15 and no more than 60.
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Sec 3.1 - 36
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
A rectangle must have an area of at least 15 cm2 and no
more than 60 cm2. If the width of the rectangle is 3 cm,
what is the range of values for the length?
Step 4
Solve.
Step 5
State the answer. In order for the rectangle to
have an area of at least 15 cm2 and no more
than 60 cm2 when the width is 3 cm, the length
must be at least 5 cm and no more than 20 cm.
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Sec 3.1 - 37
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
A rectangle must have an area of at least 15 cm2 and no
more than 60 cm2. If the width of the rectangle is 3 cm,
what is the range of values for the length?
Step 6
Check. If the length is 5 cm, the area will be
3 • 5 = 15 cm2; if the length is 20 cm, the
area will be 3 • 20 = 60 cm2. Any length
between 5 and 20 cm will produce an area
between 15 and 60 cm2.
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Sec 3.1 - 38
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
You have just purchased a new cell phone. According to
the terms of your agreement, you pay a flat fee of $6 per
month, plus 4 cents per minute for calls. If you want your
total bill to be no more than $10 for the month, how many
minutes can you use?
Step 1
Read the problem.
Step 2
Assign a variable. Let x = the number of
minutes used during the month.
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Sec 3.1 - 39
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
You have just purchased a new cell phone. According to
the terms of your agreement, you pay a flat fee of $6 per
month, plus 4 cents per minute for calls. If you want your
total bill to be no more than $10 for the month, how many
minutes can you use?
Step 3
Write an inequality. You must pay a total
of $6, plus 4 cents per minute. This total
must be less than or equal to $10.
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Sec 3.1 - 40
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
Step 4
Solve.
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Sec 3.1 - 41
3.1 Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
Step 5
State the answer. If you use no more
than 100 minutes of cell phone time, your
bill will be less than or equal to $10.
Step 6
Check. If you use 100 minutes, you will
have a total bill of $10, or $6 + $0.04(100).
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Sec 3.1 - 42