Solving Simple Inequalities

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1-5
Solving Simple Inequalities
Warm Up
Problem of the Day
Lesson Presentation
Course 3
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Warm Up Solve. 1. x 6 13 2. 8n 48 3. t ?
2 56 4. 6
x 7
n 6
t 58
z 36
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Problem of the Day Bill and Brad are taking
drivers education. Bill drives with his
instructor for one and a half hours three times a
week. He needs a total of 27 hours. Brad drives
two times a week, two hours each time. He needs
26 hours. Who will finish his hours first?
Bill
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Learn to solve and graph inequalities.
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Vocabulary
inequality algebraic inequality solution of an
inequality solution set
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An inequality compares two quantities and
typically uses one of these symbols
?
lt
is greater than
is less than
?
?
is greater than or equal to
is less than or equal to
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Additional Example 1 Completing an Inequality
Compare. Write lt or gt.
gt
lt
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Try This Example 1
Compare. Write lt or gt.
lt
gt
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An inequality that contains a variable is an
algebraic inequality.
A number that makes an inequality true is a
solution of the inequality.
The set of all solutions is called the solution
set. The solution set can be shown by graphing it
on a number line.
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x is less than 5
x lt 5
x 4
4 lt 5
x 2.1
2.1 lt 5
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a is greater than 0 a is more than 0
a gt 0
a 7
7 gt 0
a 25
25 gt 0
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y is less than or equal to 2 y is at most 2
y ? 2
y 0
0 ? 2
y 1.5
1.5 ? 2
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m is greater than or equal to 3 m is at least 3
m ? 3
m 17
17 ? 3
m 3
3 ? 3
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Most inequalities can be solved the same way
equations are solved.
Use inverse operations on both sides of the
inequality to isolate the variable.
There are special rules when multiplying or
dividing by a negative number, which you will
learn in the next chapter.
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Additional Example 2A Solving and Graphing
Inequalities
Solve and graph the inequality.
A. x 2.5 ? 8
2.5
2.5
Subtract 2.5 from both sides.
x ? 5.5
According to the graph, 5.4 is a solution, since
5.4 lt 5.5, and 6 should not be solution because 6
gt 5.5.
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Additional Example 2B Solving and Graphing
Inequalities
Solve and graph the inequality.
B. 5t gt 15
5t gt 15
Divide both sides by 5.
5
5
t gt 3
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Additional Example 2C Solving and Graphing
Inequalities
Solve and graph the inequality.
C. w 1 lt 8
1 1
Add 1 to both sides.
w lt 9
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Additional Example 2D Solving and Graphing
Inequalities
Solve and graph the inequality.
D. 3 ?
Multiply both sides by 4.
4
4
12 ? p
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Try This Examples 2A and 2B
Solve and graph each inequality.
A. x 2 ? 3.5
2
2
Subtract 2 from both sides.
x ? 1.5
B. 6u gt 72
6u gt 72
Divide both sides by 6.
6
6
u gt 12
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Try This Examples 2C and 2D
Solve and graph each inequality.
C. z 6 lt 15
6 6
Add 6 to both sides.
z lt 21
Multiply both sides by 9.
9
9
18 ? b
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Additional Example 3 Problem
Solving Application
An interior designer is planning to place a
wallpaper border along the edges of all four
walls of a room. The total distance around the
room is 88 feet. The border comes in packages of
16 feet. What is the least number of packages
that must be purchased to be sure that there is
enough border to complete the room?
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Additional Example 3 Continued
The answer will be the least number of packages
of border needed to wallpaper a room.
List the important information

• The total distance around the room is 88 feet.

• The border comes in packages of 16 feet.

Show the relationship of the information
the length of one package of border
the number of packages of border
?

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Additional Example 3 Continued
Use the relationship to write an inequality. Let
x represent the number of packages of border.
?

x
16 ft
88 feet
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Additional Example 3 Continued
16x ? 88
16x ? 88
Divide both sides by 16.
16
16
x ? 5.5
At least 5.5 packages of border must be used to
complete the room.
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Additional Example 3 Continued
Look Back
Because whole packages of border must be
purchased, at least 6 packages of border must be
purchased to ensure that there is enough to
complete the room.
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Try This Example 3
The answer will be the number of packages of
cookies a customer needs to purchase.
List the important information

• Cookies are sold in packages of 20 cookies.

• A customer needs to purchase 130 cookies.

Show the relationship of the information
the number of cookies in one package
the number of packages of cookies to be purchased
130 cookies
?

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Try This Example 3 Continued
Use the relationship to write an inequality. Let
x represent the number of packages of cookies.
?
20 cookies
130 cookies

x
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Try This Example 3 Continued
20x ? 130
20x ? 130
Divide both sides by 20.
20
20
x ? 6.5
At least 6.5 packages of cookies need to be
purchased.
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Try This Example 3 Continued
Look Back
Because whole packages of cookies must be
purchased, at least 7 packages of cookies must be
purchased for the party.
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Lesson Quiz
Use lt or gt to compare each inequality. 1. 13
5(2) 2. 14 2 11 Solve and graph each
inequality. 3. k 9 lt 12 4. 3 ? 5. A school
bus can hold 64 passengers. Three classes would
like to use the bus for a field trip. Each class
has 21 students. Write and solve an inequality to
determine whether all three classes will fit on
the bus.
gt
gt
klt 3
6 ? m