1­4 Inequalities 2010
August 27, 2010
1-4 Solving Inequalities
!
!
!
t
ee
w
s
is >
h
t
a
M
Objective:
To solve and graph inequalities
To solve and graph compound inequalities
Jun 27­1:11 PM
1
1­4 Inequalities 2010
August 27, 2010
Check Skills You'll Need
State whether each inequality is true or false.
1. 5 < 12
_
4. 5 < ­12
2. 5 < ­12
_
5. 5 < 5
_
3. 5 > 12
_
6. 5 > 5
Solve each equation.
7. 3x + 3 = 2x ­ 3
8. 5x = 9(x ­ 8) + 12
Jun 27­1:11 PM
2
1­4 Inequalities 2010
August 27, 2010
Properties of Inequalities
Let a, b, and c represent real numbers
Transitive Property
_
_
_
If a < b and b < c, then a < c
Addition Property
_
_
If a < b, then a + c < b + c
Subtraction Property
_
_
If a < b, then a ­ c < b ­ c
Multiplication Property
_
_
If a < b and c > 0, then ac < bc
_
_
If a < b and c < 0, then ac > bc
Division Property
You must reverse the inequality symbol when c is negative
a_ _ _
b
_
If a < b and c > 0, then < c c
_
a _ _b
_
If a < b and c < 0, then > c c
Jun 27­1:11 PM
3
1­4 Inequalities 2010
August 27, 2010
Solving and Graphing Inequalities
Solve each inequality. Graph the solution.
­2x < 3(x ­ 5)
­2x < 3x ­ 15
­5x < ­15
x > 3
­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5
Jun 27­1:11 PM
4
1­4 Inequalities 2010
August 27, 2010
No solutions or all real numbers as solutions
Solve each inequality. Graph the solution.
7x > 7(2 + x)
7x > 14 + 7x
0 > 14
0 > 14
/
Therefore there is no solution
­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5
Jun 27­1:11 PM
5
1­4 Inequalities 2010
August 27, 2010
Aug 27­10:47 AM
6
1­4 Inequalities 2010
August 27, 2010
Real ­ World Connection
A real estate agent earns a salary of $2000 per month plus 4% of the sales. Find the sales if the salesperson is to have a monthly income of at least $5000. Relate:
_
$2000 + 4% of sales > $5000
Define:
Let x = monthly real estate sales
Write:
_
2000 + .04x > 5000
_
.04x > 3000
_
x > 75000
Jun 27­1:11 PM
7
1­4 Inequalities 2010
August 27, 2010
Compound Inequality
A compound inequality is a pair of inequalities joined by and or or.
Examples:
­1 < x and x < 3, which you can also write as ­1 < x < 3
x < ­1 or x > 3
Jun 27­1:11 PM
8
1­4 Inequalities 2010
August 27, 2010
Compound inequality containing and
_
Graph the solution of 2x ­ 1 < 3x and x > 4x ­ 9
_
2x ­ 1 < 3x
and
x > 4x ­ 9
_
­ 1 < x
­3x > ­9
x < 3
­10 ­9
­8 ­7
­6
­5 ­4
­3
­2
­1
0
1
2
3
4
5
6
7
8
9
10
Jun 27­1:11 PM
9
1­4 Inequalities 2010
August 27, 2010
Compound inequality containing or
Graph the solution of 3x + 9 < ­3 or ­2x + 1 < 5
or
3x + 9 < ­3
­2x + 1 < 5
­2x < 4
3x < ­12
x > ­2
x < ­4
­10 ­9
­8 ­7
­6
­5 ­4
­3
­2
­1
0
1
2
3
4
5
6
7
8
9
10
Jun 27­1:11 PM
10
1­4 Inequalities 2010
August 27, 2010
_
+
A strip of wood is to be 17 cm long with a tolerance of 0.15 cm. How much should be trimmed from a strip 18 cm long to allow it to meet specifications?
Relate: _
_
minimum length < final length < maximum length
Define:
Let x = number of centimeters to remove
Write:
_
_
17 ­ 0.15 < 18 ­ x < 17 + 0.15
_
_
16.85 < 18 ­ x < 17.15
_
_
­1.15 < ­ x < ­0.85
_ _
1.15 > x > 0.85
You can trim at least 0.85 cm and no more than 1.15 cm.
Jun 27­1:11 PM
11
1­4 Inequalities 2010
August 27, 2010
What is one important difference between solving equations and solving inequalities?
Jun 27­1:11 PM
12
1­4 Inequalities 2010
August 27, 2010
Homework
pages: 29 ­ 30 #'s: 2 ­ 28 even, 29 ­ 32, 36 ­ 37, 42 ­ 54 even
Jun 27­1:11 PM
13