1. Inequalities and Applications
Section 9.1
Inequalities and Applications

2. Recall
Solve and check the equation
3x -2 (x + 3) = 2(x - 3)

3. Properties to Solve Inequalities
The Addition Principle for Inequalities
For any real number a, b, and c
a < b is equivalent to a + c < b + c
a > b is equivalent to a + c > b + c
The Multiplication Principle for Inequalities
For any real numbers a, b, and for any positive number c,
a < b is equivalent to ac < bc
a > b is equivalent to ac > bc
For any real numbers a, b, and for any negative number c,
a < b is equivalent to ac > bc
a > b is equivalent to ac < bc

4. Types of answer No Solution All Real Solution Group of Solutions
The variable cancels out and you have a false statement
All Real Solution
The variable cancels out and you have a true statement
Group of Solutions
The variable does not cancel out and you have three ways to write the answer

5. How to Write the Answer Set Builder Notation Interval Notation
{variable | interval }
Interval Notation
[included, included],
[included, not included),
(not, included, included],
(not included, not included)
Graphing Notation
Use open circle or parentheses to not include the value
Use filled in circle or bracket to include the value

6. Example
Solve the inequality and write answer in set builder notation 2x – 5 ≥ 9

7. Example
Solve the inequality and write answer in interval notation form. 8 – x < 15

8. Example
Solve the inequality and write answer in graphing notation 2(4 + 2x) > 2x + 3(2 - 5x)

9. Example
Solve the inequality and write the answer in all three forms. 3(x – 2) + 3 < 2(x – 1) + x

10. Example
Solve the inequality and write the answer in all three forms. 3x – 2 < 2(x – 1) + x

11. Word Problem
Jenn can rent a moving truck for either $99 with unlimited mileage or $49 plus $0.80 per mile. For what mileages would the unlimited mileage plan save money?

12. Homework
9.1 11, 17, 19, 21, 23, 39, 41, 45