1. 1. > < ≥ ≤ Signs of Inequality Term Definition Example
Greater than
x > 2
x is greater than 2
<
Less than
x < 1
x is less than 1 ≥
Greater than or equal to
x ≥ 2
x is greater than or equal to 2 ≤
Less than or equal to
x ≤ 1
x is less than or equal to 1
On a number line:
Open Circles mean > or <
Shaded Circles
mean ≥ or ≤

2. 2. Graphing Inequalities greater than X is ____________ -1
Simple Inequalities Examples
greater than
a) X > -1
X is ____________ -1
less than
X is ____________ 3
b) X < 3
greater than
or equal to
c) X ≥ 1
X is ____________ 1

3. 2. Graphing Inequalities b) -4 < a ≤ 4
Compound Inequalities Examples
b) -4 < a ≤ 4
c) a ≤ 1 OR a ≥ 3
d) -2 ≤ c < 3

4. 3. Linear inequalities are inequalities with more than one term
Solving Linear Inequalities
Linear inequalities are inequalities with more than one term
Ex: 2x + 5 > 11
We solve linear equalities exactly the same as linear equations EXCEPT
The direction of the inequality changes when multiplying or dividing by negative numbers

5. 3. Ex 1: -3x + 5 > 26 - 5 -5 -3x > 21 ÷ -3 ÷ -3 x < -7
Solving Linear Inequalities
Examples
Ex 1: x + 5 > 26
-3x > 21
÷ ÷ -3
THE INEQUALITY CHANGED DIRECTION
x < -7
Check: try x = -10
-3(-10) + 5 = 30+5 = 35 > 26
CHECK!

6. 3.
Solving Linear Inequalities
Examples
Ex 2:
2x ≥ 4x + 20
- 4x x
-2x ≥
-2x ≥
÷ ÷ -2
x ≤ -5
THE INEQUALITY CHANGED DIRECTION
Check: try x = -10
2(-10) + 10 ≥ 4(-10) + 20
-10 ≥ -20 CHECK!

7. Example 1 3.
Solving Linear Inequalities
Word Problems
Examples
Your cell phone plan costs $35 plus $0.05 per text message. What is the maximum number of messages, t, you can send to spend less than $50?
What are we trying to find?
Maximum number of messages for bill to cost less than $50
What do we know?
t = text messages
$35 plus $0.05 per text
Total cost < $50

8. 3. 35 + .05t < 50 -35 -35 .05t < 15 ÷ .05 ÷ .05 t < 300
Example 1 3.
Solving Linear Inequalities
Word Problems
Examples
Your cell phone plan costs $35 plus $0.05 per text message. What is the maximum number of messages, t, you can send to spend less than $50?
How do we solve it?
Did we answer the original question?
t < 50
Check: try t = 299
(299) = < 50
CHECK!
.05t < 15
÷ ÷ .05
t < 300
300 messages

9. 3. X Y -3 2 -1 Ex 1: Graph this linear inequality: Y > x - 3
Graphing Linear Inequalities
2 Variables!
Ex 1: Graph this linear inequality: Y > x - 3 X Y
Treat it as an equation and graph the line by plotting two ordered pairs
The line is solid if “or equal to” and dotted if just greater or less than
Read from Y perspective
Shade under if less than
Shade over if greater than -3 2 -1
TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

10. 3. X Y 5 2 3 Ex 2: Graph this linear inequality: Y ≤ -x + 5
Graphing Linear Inequalities
2 Variables!
Ex 2: Graph this linear inequality: Y ≤ -x + 5
Treat it as an equation and graph the line by plotting two ordered pairs
The line is solid if “or equal to” and dotted if just greater or less than
Start where x is
Shade under if less than
Shade over if greater than X Y 5 2 3
TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

11. 3. X Y 6 2 2 Ex 3: Graph this linear inequality: 2x + y > 6
Graphing Linear Inequalities
2 Variables!
Ex 3: Graph this linear inequality: 2x + y > 6 X Y
Treat it as an equation and graph the line by plotting two ordered pairs
The line is solid if “or equal to” and dotted if just greater or less than
Read from Y perspective
Shade under if less than
Shade over if greater than 6 2 2
TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!

12. 4. X Y X Y -3 5 2 -1 2 3 Ex 1: Graph this linear system: Y > x – 3
Graphing
Linear Systems
Y > x - 3
Y ≤ -x + 5 X Y X Y -3 5 2 -1 2 3
TEST AN ORDERED PAIR IN THE SHADED AREA TO CHECK!!!
Graph each inequality
The common shaded area is the solution