Transcript Solving Linear Inequalities in One Variable

6.1 Solving Linear Inequalities in One
Variable
A linear inequality in one variable is an inequality
which can be put into the form
ax + b > c
where a, b, and c are real numbers.
Note that the “>” can be replaced by , <, or .
Examples: Linear inequalities in one variable.
2x + 3 > 4
2x – 2 < 6x – 5 can be written –4x + (–2) < –5.
6x + 1  3(x – 5) can be written 6x + 1  –15.
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6.1 Solving Linear Inequalities in One
Variable
The solution set for an inequality can be expressed in two ways.
Example: Express the solution set of x < 3 in two ways.
Open circle indicate that the
number is not included in the
solution set.
1. Set-builder notation: {x | x < 3}
2. Graph on the real line:
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Example: Express the solution set of x  4 in two ways.
1. Set-builder notation: {x | x  4}
2. Graph on the real line:
Closed circle indicate that
the number is included in
the solution set.
-4 -3 -2 -1
0
1
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4
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6.1 Solving Linear Inequalities in One
Variable
A solution of an inequality in one variable is a number
which, when substituted for the variable, results in a true
inequality.
Examples: Are any of the values of x given below
solutions of 2x > 5?
?
?
x=2
2(2) > 5
4 > 5 False 2 is not a solution.
?
?
x = 2.6 2(2.6) > 5 5.2 > 5 True 2.6 is a solution.
?
?
x=3
2(3) > 5
6 > 5 True 3 is a solution.
?
?
x = 1.5 2(1.5) > 5
3 > 5 False 1.5 is not a solution.
The solution set of an inequality is the set of all solutions.
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6.1 Solving Linear Inequalities in One
Variable
The graph of a linear inequality in one variable is the graph
on the real number line of all solutions of the inequality.
Verbal Phrase
Inequality
All real numbers less than 2
x<2
All real numbers greater than -1
x > -1
All real numbers less than or
equal to 4
x<4
All real numbers greater than
or equal to -3
x > -3
Graph
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6.1 Solving Linear Inequalities in One
Variable
Addition and Subtraction Properties
• If a > b and c is a real number, then
a > b, a + c > b + c, and a – c > b – c have the same solution set.
• If a < b and c is a real number, then
a < b, a + c < b + c, and a – c < b – c have the same solution set.
Example: Solve x – 4 > 7.
x–4+4>7+4
x > 11
{x | x > 11}
Add 4 to each side of the inequality.
Set-builder notation.
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6.1 Solving Linear Inequalities in One
Variable
Example: Solve 3x  2x + 5.
3x – 2x  2x + 5 – 2x Subtract 2x from each side.
x5
Set-builder notation.
{x | x  5}
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6.1 Solving Linear Inequalities in One
Variable
Multiplication and Division Properties
• If c > 0 the inequalities a > b, ac > bc, and
same solution set.
• If c < 0 the inequalities a > b, ac < bc, and
same solution set.
Example: Solve 4x  12.
4x  12
( 4) ( 4)
x3
a
>
c
a
<
c
b
have the
c
b
have the
c
Divide by 4.
4 is greater than 0, so the
inequality sign remains the same.
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6.1 Solving Linear Inequalities in One
Variable
1
Example: Solve  x  4 .
3
1
 x  (– 3)  4  (–3)
3
x  12
Multiply by – 3.
– 3 is less than 0, so the inequality sign
changes.
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6.1 Solving Linear Inequalities in One
Variable
Example: Solve x + 5 < 9x + 1.
– 8x + 5 < 1
– 8x < – 4
1
2
1

x
|
x



2


x>
Subtract 9x from both sides.
Subtract 5 from both sides.
Divide both sides by – 8 and simplify.
Inequality sign changes because of
division by a negative number.
Solution set in set-builder notation.
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6.1 Solving Linear Inequalities in One
Variable
Example: Solve 4 x  2  3 x  4.
5
5
1
x2 4
5
1
x6
5
x  30
•
Subtract
3
x from both sides.
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Add 2 to both sides.
Multiply both sides by 5.
Solution set as a graph.
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6.2 Problem Solving
Example: A cell phone company offers its customers a rate of
$89 per month for 350 minutes, or a rate of $40 per month
plus $0.50 for each minute used.
How many minutes per month can a customer who chooses the
second plan use before the charges exceed those of the first plan?
Let x = the number of minutes used.
Solve the inequality 0.50x + 40  89 .
0.50x  49
Subtract 40.
x  24.5 Divide by 0.5.
The customer can use up to 24.5 minutes per month before the
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cost of the second plan exceeds the cost of the first plan.
6.3 Compound Inequalities
A compound inequality is formed by joining two inequalities
with “and” or “or.”
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6.3 Compound Inequalities
Example: Solve x + 2 < 5 and 2x – 6 > – 8.
Solve the first inequality.
Solve the second inequality.
x+2<5
2x – 6 > – 8
x < 3 Subtract 2.
2x > – 2 Add 6.
{x | x < 3} Solution set
x > – 1 Divide by 2.
{x | x > –1} Solution set
The solution set of the “and” compound inequality is the
intersection of the two solution sets.
x | x  3 x | x  1 x | 1  x  3
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6.3 Compound Inequalities
When solving compound inequalities, it is possible to work
with both inequalities at once.
This inequality means
Example: Solve 11 < 6x + 5 < 29. 11 < 6x + 5 and 6x + 5 < 29.
6 < 6x < 24 Subtract 5 from each of the three parts.
1 < x < 4 Divide 6 into each of the three parts.
x | 1  x  4 Solution set.
-4 -3 -2 -1
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1
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4
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6.3 Compound Inequalities
1
Example: Solve 8   x  6  5.
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1
2   x  1 Subtract 6 from each part.
2
Multiply each part by – 2.
Multiplication by a negative number
4 x  2
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changes the inequality sign for each part.
Solution set.
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0
1
2
3
4
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6.3 Compound Inequalities
Example: Solve x + 5 > 6 or 2x < – 4.
Solve the first inequality.
Solve the second inequality.
2x < – 4
x < –2
{ x | x < – 2} Solution set
x+5>6
x>1
{ x | x > 1} Solution set
Since the inequalities are joined by “or” the solution set is the
union of the solution sets.
x | x  1 x | x  2
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6.4 Absolute Value and Inequalities
There are 4 types of absolute value
inequalities and equivalent inequalities




|x| <
|x| <
|x| >
|x| >
a
a
a
a
a  x  a
a  x  a
x  a or x  a
x   a or x  a
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6.4 Absolute Value and Inequalities
Translating Absolute Value Inequalities
1. The inequality |ax + b| < c is
equivalent to -c < ax + b < c
2. The inequality |ax + b| > c is
equivalent to ax + b < -c or
ax + b > c
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6.4 Absolute Value and Inequalities
Example Solve |x - 4| < 3
 -3 < x - 4 < 3
 1 < x < 7
 The solution set is {x| 1 < x < 7} and the
interval is (1, 7)
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6.4 Absolute Value and Inequalities
Example Solve |4x - 1| < 9
 -9 ≤ 4x - 1 ≤ 9
 -8 ≤ 4x ≤ 10
 -2 ≤ x ≤ 5/2
 The interval solution is [-2, 5/2]

-4 -3 -2 -1

0
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6.4 Absolute Value and Inequalities
Solve |x| > a
 Solve |x + 1| > 2
 x + 1 < -2 or x + 1 > 2
 x < -3 or x > 1
 The solution interval is (-∞, -3) U (1, ∞)
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6.4 Absolute Value and Inequalities





Solve |x| ≥ a
Solve |2x - 8| ≥ 4
2x – 8 ≤ -4 or 2x - 8 ≥ 4
2x ≤ 4 or 2x ≥ 12
x ≤ 2 or x ≥ 6
The solution interval is (-∞, 2] U [ 6, ∞)
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6.5 Linear Inequalities

A linear inequality in two variables can
be written in any one of these forms:





Ax + By < C
Ax + By > C
Ax + By ≤ C
Ax + By ≥ C
An ordered pair (x, y) is a solution of the
linear inequality if the inequality is
TRUE when x and y are substituted into
the inequality.
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6.5 Linear Inequalities

Which ordered pair is a solution of
5x - 2y ≤ 6?
A.
B.
C.
D.
(0, -3)
(5, 5)
(1, -2)
(3, 3)
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6.5 Linear Inequalities

The graph of a linear inequality is
the set of all points in a coordinate
plane that represent solutions of the
inequality.

We represent the boundary line of the
inequality by drawing the function
represented in the inequality.
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6.5 Linear Inequalities

The boundary line will be a:
Solid line when ≤ and ≥ are used.
 Dashed line when < and > are
used.


Our graph will be shaded on one
side of the boundary line to show
where the solutions of the
inequality are located.
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6.5 Linear Inequalities
Here are some steps to help graph linear
inequalities:
Graph the boundary line for the inequality.
Remember:
1.


≤ and ≥ will use a solid curve.
< and > will use a dashed curve.
Test a point NOT on the boundary line to determine
which side of the line includes the solutions. (The
origin is always an easy point to test, but make sure
your line does not pass through the origin)
2.


If your test point is a solution (makes a TRUE statement),
shade THAT side of the boundary line.
If your test points is NOT a solution (makes a FALSE
statement), shade the opposite side of the boundary line.
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6.5 Linear Inequalities




Graph the inequality x ≤ 4 in a coordinate
plane
Decide whether to
use a solid or
dashed line.
Use (0, 0) as a
test point.
Shade where the
solutions will be.
y
5
x
-5
-5
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6.5 Linear Inequalities





Graph y > x + 2 in a coordinate plane.
Sketch the boundary line of the graph.
Solid or dashed
line?
Use (0, 0) as a
test point.
Shade where the
solutions are.
y
5
x
-5
-5
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6.5 Linear Inequalities





Graph y > -½x - 2 in a coordinate plane.
Sketch the boundary line of the graph.
Solid or dashed
line?
Use (0, 0) as a
test point.
Shade where the
solutions are.
y
5
x
-5
-5
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6.5 Linear Inequalities


Graph 3x - 4y > 12 in a coordinate plane.
Sketch the boundary line of the graph.




Find the x- and
y-intercepts and
plot them.
Solid or dashed
line?
Use (0, 0) as a
test point.
Shade where the
solutions are.
y
5
x
-5
-5
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Example 4:
Using a new Test Point


Graph y < 2/5x in a coordinate plane.
Sketch the boundary line of the graph.


Find the x- and y-intercept and plot them.y
Both are the origin! 5
• Use the line’s slope



to graph another point.
Solid or dashed
line?
Use a test point
OTHER than the
origin.
Shade where the
solutions are.
-5
-5
x
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