Transcript 2.5 Linear Inequalities in One Variable

2.5
Linear Inequalities in One Variable
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 3.1 - 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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
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.
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Linear Inequalities in One Variable
Linear Inequality
An inequality says that two expressions are not equal.
Linear Inequality
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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3.1 Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Linear Inequalities in One Variable
Using the Addition Property of Inequality
Solve and graph the solution:
Linear Inequalities in One Variable
Multiplication Property of Inequality
Multiplication Property of Inequality
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Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
Linear Inequalities in One Variable
Using the Multiplication Property of Inequality
Solve and graph the solution:
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 - 12
Linear Inequalities in One Variable
Solving a Linear Inequality
Solve and graph the solution:
Linear Inequalities in One Variable
Solving a Linear Inequality with Fractions
Solve and graph the solution:
Linear Inequalities in One Variable
Solving a Three-Part Inequality
Solve and graph the solution:
Linear Inequalities in One Variable
Solving a Three-Part Inequality
Solve and graph the solution:
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
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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Interpretation
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
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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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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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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Linear Inequalities in One Variable
Solving Applied Problems Using Linear Inequalities
Step 4
Solve.
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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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