1. 2.6 Solving Linear Inequalities

2. Inequality Graph Interval Notation
x > 2 | | | ( | | (2, )
x < | | | ) | | (-, 2)
x ≥ | | | [ | | [2, )
x ≤ | | | ] | | (-, 2]

3. If the inequality symbol is > or < use ( or )
Note:
If the inequality symbol is > or < use ( or )
the solution number is not included in the solution set
If the inequality symbol is ≥ or ≤ use [ or ]
the solution number IS included in the solution set

4. Addition Property of Inequalities
Adding or subtracting the same quantity from each side of an inequality will not change the solution set.
EX: < < 9
3 + 1 < – 1 < 9 – 1
4 < < 8
True True

5. Ex: Solve and graph the solution set:
x + 5 > 12
x + 5 – 5 > 12 – 5
x > 7
| | | ( | |
Solution: (7, )

6. Multiplication Property of Inequalities
Multiplying/Dividing both sides of an inequality by a POSITIVE number does NOT change the solution set
Mult/Div both sides of an inequality by a NEGATIVE number REQUIRES the inequality symbol to be REVERSED to produce an equivalent inequality
REVERSE SYMBOL
EX: < < < 9
3 ۰ 1 < 9 ۰ ۰ (–1) < 9۰ (–1) ۰ (–1) > 9۰ (–1)
3 < –3 < – –3 > –9
True False True

7. Ex: Solve and graph the solution set
Ex: Solve and graph the solution set. 7y < y < 2 | | ) | | | Solution: (-, 2)

8. Ex: Solve and graph the solution set
Ex: Solve and graph the solution set. -3a < 12 -3a > divide by neg. # so reverse ineq. sym a > -4 | ( | | | | Solution: (-4, )

9. Ex: Solve and graph the solution set
Ex: Solve and graph the solution set. -5x + 6 ≤ -9 -5x + 6 – 6 ≤ -9 – 6 -5x ≤ x ≥ divide by neg. # so reverse ineq. sym x ≥ 3 | | | [ | | Solution: [3, )

10. Unusual Stuff
Ex: Solve . 3x – 5 < 3(x – 2) 3x – 5 < 3x – 6 3x – 5 – 3x < 3x – 6 – 3x -5 < -6 No variables remain and stmt. is FALSE, so no soln. Answer: ø

11. Ex: Solve. 5(x + 4) > 5x x + 20 > 5x x + 20 – 5x > 5x + 10 – 5x 20 > 10 No variables remain and stmt. is TRUE, so all real #s are solns. Answer: (- , )

12. Application Problems
WORDS SYMBOLS x is at least 12 x ≥ 12 x is more than 12 x > 12 x is at most 13 x ≤ 13 x is less than 13 x < 13

13. Ex. A car can be rented from Continental Rental for $80 per week plus 25 cents for each mile driven. How many miles can you travel if you can spend at most $400 for the week? x = # miles x ≤ x – 80 ≤ 400 – x ≤ x ≤ 1280
You can drive up to 1,280 miles