Download IntAlgebra 1-4 Inequalities

Inequalities and Their
Graphs
Inequalities –
What do they mean in words?
•Less than or smaller than
•Fewer than
•Less than or equal to •No more than
•At most
•A maximum of
•Greater than or bigger than
•More than
•Greater than or equal to •No less than
•At least
•A minimum of
Example:
Write an inequality to describe people who are
NOT of legal driving age in California
First- try to put it in words
Who are the people who can not drive?
Everyone younger than or less than 16 years old
people who can’t drive are less than 16
p<16
When we start graphing we have to remember…
Use OPEN bullet for
Use CLOSED bullet for
1. Graph X  -1

2. Graph K >

-2
-1
0
1
2
3
-2
-1
0
1
2
3
X
2
3. Write the inequality
-2
-1
0
1
2
3
Try this one on your own or with your partner
Solve  y     y   AND graph it
 y     y  
-7y
-7y
4 > y - 10
+10
+10
14 > y
12
13
14
15
16
17
Check out what happens with
Division AND Multiplication
12 > 4
THIS IS TRUE
12 > 4
4
-4 -4
3>1
-3 > -1
4
THIS IS STILL TRUE
THIS IS TRUE
NOT TRUE
So something weird happens when we divide by a negative
THE RULES
If you divide OR multiply by a
negative number you must “flip”
the sign
>


Example
3x  27
3 3
x

 -9
We divided by a
negative we have to
FLIP
Solve & Graph each Inequality
4(x  3)  7  4x 1
No Solution – It doesn’t make sense!
Solve & Graph each Inequality
2x  2(x 1)  3
ALL real numbers are solutions!
Writing Compound Inequalities
x is greater than –4 and less than or equal to –2.

x 44
4xand
x x2 2
l
-6
l
l
-5
l
-4
l
l
-3
l
-2
-1
0
Writing Compound Inequalities
x is greater than 3 or less than –1.
x  3 or x  1
l
-2
l
l
-1
l
0
l
l
1
l
2
3
4
Solving a Compound Inequality with And
Solve the inequality and graph the
solution.
3 2x  1 7
1
1 1
4  2x  6
2 2
2
2 x  3

l
-3
l
-2
l
l
-1
l
0
l
l
1
l
2
3
Solving a Compound Inequality with And
Solve the inequality and graph the
solution.
4.  6  3  x  4
3 3
3
3  x  1
l
-5
l
-4
l
l
-3
l
-2
l
l
-1
l
0
1
Solving a Compound Inequality with And
Solve the inequality and graph the
solution.
5. 2  3x  8  17
8
8 8
6  3x  9
3 3 3
2  x  3
3  x  2
l
-4
l
-3
l
l
-2
l
-1
l
l
0
l
1
2
Solving a Compound Inequality with Or
Solve the inequality and graph the
solution.
6. 2 x  3  5 or 3x  1  16
3 3
2x  8
1 1
2 2
x 4
l
3
l
l
l
4
l
or
l
l
5
3x  15
3 3
x5
l
6
Solving a Compound Inequality with Or
Solve the inequality and graph the
solution.
7.  4 x  2  6 or 2 x  6
2 2
2 2
4x  4
4 4
x  1 or x  3
l
l
-6 -5
l
l
-4
l
-3
l
l
-2
l
-1
0
Writing and Using a Linear Model
In 1985, a real estate property was sold for $172,000.
The property was sold again in 1999 for $226,000.
Write a compound inequality that represents the
different values that the property was worth between
1995 and 1999.
172
x ,172
000,000
 x and x  226,000
172,000  x  226,000