Transcript Chapter Two 2.5

Section 2.5
Linear
Inequalities
Copyright © 2013 Pearson Education, Inc.
Page 136
Solutions and Number Line Graphs
A linear inequality results whenever the equals sign in a
linear equation is replaced with any one of the symbols <,
≤, >, or ≥.
x > 5,
3x + 4 < 0, 1 – y ≥ 9
A solution to an inequality is a value of the variable that
makes the statement true. The set of all solutions is
called the solution set.
Example
Page 137
Use a number line to graph the solution set to each
inequality.
a. x  1
b. x  1
c. x  3
d. x  2
Linear Inequalities in One Variable
Page 137
)
a. x  3
b. x  1
[
c.  1  x  3
(
]
Interval Notation
Page 137
Each number line graphed on the previous slide
represents an interval of real numbers that corresponds
to the solution set to an inequality.
Brackets and parentheses can be used to represent the
interval.
For example:
x  1  (1, )
x  1  [1, )
Example Interval Notation
Page 137
Write the solution set to each inequality in interval
notation.
y  2
a.
b.
x6
Solution
a.
(6, )
b.
More examples
c. x  0
[0, )
d. x  5
(,5)
(, 2]
Example Checking a Solution
Page 138
Determine whether the given value of x is a solution to
the inequality. 4 x  2  8,
x7
Solution
?
4x  2 < 8
?
4(7)  2  8
?
28  2  8
?
26  8 False
The Addition Property of Inequalities
Page 139
Example
Page 139
Solve each inequality. Then graph the solution set.
a. x – 2 > 3
b. 4 + 2x ≤ 6 + x
Solution
a.
x–2>3
x–2+2>3+2
x>5
b. 4 + 2x ≤ 6 + x
4 + 2x – x ≤ 6 + x – x
4+x≤6
4–4+x≤6–4
x≤2
Properties of Inequalities
Page 140
)
x69
6 6
(,3)
x3
{x | x  3}
8x  2  7 x  4
2
2
8x  7 x  2
- 7x
- 7x
x  2
[
[ 2, )
{x | x  2}
The Multiplication Property of Inequalities
Page 141
Example
Page 141
Solve each inequality. Then graph the solution set.
a. 4x > 12
b. 2   1 x
4
Solution
1
a. 4x > 12
b. 2   x
4 x 12

4
4
x3
4
1
4( 2)  ( 4)  x
4
8 x
Example
Page 142
Solve each inequality. Write the solution set in
set-builder notation.
a. 4x – 8 > 12
b. 4  3 x  4 x  5
Solution
4  3 x  4 x  5
b.
a. 4x – 8 > 12
4 x  8  8  12  8
4 x  20
x5
{x | x  5}
4  3x  3x  4 x  5  3x
4  x  5
4  5  x  5  5
9  x
{x | x  9}
Properties of Inequalities
)
4 1 x  2 4
4
(,8)
{x | x  8}
x8
(
 6 x  18
6 6
(3, )
Sign changes
x  3
Page 142
Linear Inequalities
6  3x  5 x  2
8  3x  5 x
Page 142
6  3x  5 x  2
 3x  5 x  8
Add 2
Add 3x
8  8x
8
x 1
Subtract 5x
 8 x  8
8
1 x
Subtract 6
-8
-8
x 1
same
[
[1,  )
Sign changes
Linear Inequalities
Page 142
2( x  3)  1  3( x  2)  14
2 x  6  1  3x  6  14
2 x  7  3x  8
2 x  1  3x
Add 8
Sub 2x
1 x
x 1
[
[1,  )
Applications
Page 143
To solve applications involving inequalities, we often
have to translate words to mathematical statements.
Example
Page 143
Translate each phrase to an inequality. Let the variable
be x.
a. A number that is more than 25.
x > 25
b. A height that is at least 42 inches.
x ≥ 42
Example
Page 144
For a snack food company, the cost to produce one case
of snacks is $135 plus a one-time fixed cost of $175,000
for research and development. The revenue received
from selling one case of snacks is $250.
a. Write a formula that gives the cost C of producing x
cases of snacks.
C = 135x + 175,000
b. Write a formula that gives the revenue R from selling x
cases of snacks.
R = 250x
Example (cont)
Page 144
For a snack food company, the cost to produce one case of snacks is $135 plus a onetime fixed cost of $175,000 for research and development. The revenue received from
selling one case of snacks is $250.
c. Profit equals revenue minus cost. Write a formula that
calculates the profit P from selling x cases of snacks.
P=R–C
= 250x – (135x + 175,000)
= 115x – 175,000
Example (cont)
Page 144
For a snack food company, the cost to produce one case of snacks is $135 plus a onetime fixed cost of $175,000 for research and development. The revenue received from
selling one case of snacks is $250.
d. How many cases need to be sold to yield a positive
profit?
115x – 175,000 > 0
115x > 175,000
x > 1521.74
Must sell at least 1522 cases.
DONE
Objectives
•
Solutions and Number Line Graphs
•
The Addition Property of Inequalities
•
The Multiplication Property of Inequalities
•
Applications
EXAMPLE
Graphing inequalities on a number line
Use a number line to graph the solution set to each inequality.
a.
b.
c.
x 1
x 1
d.
x5
x2