Transcript Ch. 2.8 power point

Chapter 2
Section 8
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2.8
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Solving Linear Inequalities
Graph intervals on a number line.
Use the addition property of inequality.
Use the multiplication property of inequality.
Solve linear inequalities by using both
properties of inequality.
Solve applied problems by using inequalities.
Solve linear inequalities with three parts.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Linear Inequalities
Inequalities are algebraic expressions related by
<
>
“is less than,”
“is greater than,”
≤ “is less than or equal to,”
≥ “is greater than or equal to.”
We solve an inequality by finding all real number solutions of
it. For example, the solution set {x | x ≤ 2} includes all real
numbers that are less than or equal to 2, not just the integers less
than or equal to 2.
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Objective 1
Graph intervals on a number
line.
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Slide 2.8 - 4
Graph intervals on a number line.
A good way to show the solution set of an inequality is by
graphing.
We graph all the real numbers belonging to the set {x | x ≤ 2}
by placing a square bracket at 2 on a number line and drawing an
arrow extending from the bracket to the left (to represent the fact
that all numbers less than 2 are also part of the graph).
Some texts use solid circles rather than square brackets to
indicate the endpoint is included in a number line graph. (Open
circles are also used to indicate noninclusion, rather than
parentheses, when the endpoint is not included in the graph.)
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Graph intervals on a number line. (cont’d)
The set of numbers less than or equal to 2 is an example of an
interval on the number line. To write intervals, we use interval
notation. For example, the interval of all numbers less than or
equal to 2 is written (−∞, 2].
The negative infinity symbol −∞ does not indicate a number,
but shows that the interval includes all real numbers less than 2.
As on the number line, the square bracket indicates that 2 is
part of the solution.
A parentheses is always used next to the infinity symbol. The
set of real numbers is written as (−∞, ∞).
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EXAMPLE 1
Graphing Intervals Written in
Interval Notation on a Number
Line
Write each inequality in interval notation and graph the
interval.
x3
Solution: [3,  )
2 x4
Solution: (2, 4]
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Objective 2
Use the addition property of
inequality.
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Use the addition property of inequality.
A linear inequality in one variable can be written in the form
Ax  B  C 
where A, B, and C are real numbers, with A ≠ 0.
Examples of linear inequalities in one variable are
x  5  2
y  3  5
2k  5  10.
Linear Inequalities
(All definitions and rules are also valid for >, ≥, and ≤.)
Consider the inequality 2 < 5. If 4 is added to each side, the
result is
2 4  5 4
6  9
a true sentence.
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Use the addition property of inequality. (cont’d)
Now subtract 8 from each side:
2 8  5 8
6  3.
The result again is a true sentence. These examples suggest the
addition property of inequality.
For any real numbers A, B, and C, the inequalities
and
AC  B C
A B
have exactly the same solutions.
That is, the same number may be added to each side of an
inequality without changing the solutions.
As with the addition property of equality, the same number
may be subtracted from each side of an inequality.
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EXAMPLE 2
Using the Addition Property
of Inequality
Solve the inequality; then graph the solution set.
1  8x  7 x  2
Solution:
1  8x  7 x  7 x  2  7x
1  x 1  2 1
x
(,3)
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Objective 3
Use the multiplication property
of inequality.
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Use the multiplication property of
inequality.
The addition property of inequality cannot be used to solve an
inequality such as 4x ≥ 28. This inequality requires the
multiplication property of inequality.
Multiply each side of the inequality 3 < 7 by the positive
number 2.
2  3  2  7 
True
6 
Now multiply by each side of 3 < 7 by the negative number −5.
5  3  5  7 
False
15  
To get a true statement when multiplying each side by −5,
we must reverse the direction of the inequality symbol.
5  3  5  7 
True
15  
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Use the multiplication property of
inequality. (cont’d)
In summary, the multiplication property of inequality has
two parts.
For any real numbers A, B, and C, with C ≠ 0,
1. if C is positive, then the inequalities
A  B and AC  BC
have exactly the same solutions;
2. if C is negative, then the inequalities
A  B and AC  BC
have exactly the same solutions.
That is, each of an inequality may be multiplied by the same
positive number without changing the solutions. If the multiplier
is negative, we must reverse the direction of the inequality symbol.
The multiplication property of inequality also permits division
of each side of an inequality by the same nonzero number.
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EXAMPLE 3
Using the Multiplication Property
of Inequality
Solve the inequality; then graph the solution set.
r  12
Solution:
r 

2
2
r 6
(, 6)
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Objective 4
Solve linear inequalities by
using both properties of
inequality.
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Slide 2.8 - 16
Solve linear inequalities by using both
properties of inequality.
To solve a linear inequality, use the following steps.
Step 1: Simplify each side separately. Use the distributive
property to clear parentheses and combine like terms
on each side as needed.
Step 2: Isolate the variable terms on one side. Use the
addition 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, where k is a number.
Remember: Reverse the direction of the inequality symbol
only when multiplying or dividing each side of an inequality by
a negative number..
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EXAMPLE 4
Solving a Linear Inequality
Solve the inequality; then graph the solution set.
Solution:
7 
 ,
3 
5x  x  2  7 x  5
4x  2  7 x  7 x  5  7 x
3x  2  2  5  2
3 x 7

3 3
7
x
3
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EXAMPLE 5
Solving a Linear Inequality
Solve the inequality; then graph the solution set.
4  x  1  3x  15   2 x  1
Solution:
4x  4  3x  2x    x  2x
3x  4  4    4
3 x 12

3
3
x  4
 4,  
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Objective 5
Solve applied problems by
using inequalities.
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Solve applied problems by using inequalities.
Inequalities can be used to solve applied problems involving
phrases that suggest inequality. The table gives some of the more
common such phrases, along with examples and translations.
We use the same six problem-solving steps from Section 2.4,
changing Step 3 to “Write an inequality” instead of “Write an
equation.”
Do not confuse statements such as “5 is more than a number” with
phrases like “5 more than a number.” The first of these is expressed
as 5 > x, while the second is expressed as x + 5 or 5 + x.
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EXAMPLE 6
Finding an Average Test Score
Maggie has scores of 98, 86, and 88 on her first
three tests in algebra. If she wants an average of at
least 90 after her fourth test, what score must she
make on that test?
Solution: Let x = Maggie’s fourth test score.
98  86  88  x
x  88
 90
4
272  x
   4 
Maggie must get greater
 4
4
than or equal to an 88.
  x  272  360  272
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Objective 6
Solve linear inequalities with
three parts.
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Solve linear inequalities with three parts.
Inequalities that say the one number is between two other
numbers are three-part inequalities. For example,
3    
says that 5 is between −3 and 7.
For some applications, it is necessary to work with a three-part
inequality such as
3 x2
where x +2 is between 3 and 8. To solve this inequality, we
subtract 2 from each of the three parts of the inequality.
3  2  x  2  2   2
1 x  
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Solve linear inequalities with three parts.
(cont’d)
The idea is to get the inequality in the form
a number < x < another number,
using “is less than.” The solution set can then easily be graphed.
When inequalities have three parts, the order of the parts is important. It
would be wrong to write an inequality as 8 < x + 2 < 3, since this would
imply 8 < 3, a false statement. In general, three-part inequalities are
written so that the symbols point in the same direction and both point
toward the lesser number.
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EXAMPLE 7
Solving Three-Part Inequalities
Solve 2  3x 1  8 and graph the solution set.
Solution:
2 1  3x 1 1  8 1
3 3x 9


3 3 3
1 x  3
1,3
Remember to work with all three parts of the inequality.
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