3. Divide both sides by 3
Add 10 to both sides
Subtract 5 from both sides
Multiple both sides by -2
Multiple both sides by 2
Divide both sides by -3

5. Example: Solve the inequality, and graph the solution.
2(x – 3) < 4x + 10
2x – 6 < 4x + 10 Distribute.
– 2x – 6 < 10 Subtract 4x from both sides.
– 2x < 16 Add 6 from both sides.
x > – 8 Divide both sides by -2, switch inequality. x

6. Interval notation, is used to write the numerical solution for inequalities.
Use a bracket if you want to include the number
Use a parenthesis if you DO NOT want to include the number.
x  7 x
Numerical Notation call “Interval Notation”:
x > -4 x
Numerical Notation call “Interval Notation”:

7. –5  x means x > –5 [–5, ∞) x < 3 x NOT included (–∞, 3)3 x
NOT included
Interval notation:
(–∞, 3)
–5  x means x > –5
Included -5 x
Interval notation:
[–5, ∞)

8. x

9. x  x – 7 Subtract 5 from both sides.
Example: Solve the inequality. Graph the solution and give your answer in interval notation.
x + 5  x – 2
x  x – 7 Subtract 5 from both sides.
0  – 7 Subtract x from both sides.
(Always true!)
Since 0 is always greater than –7, the solution is all real numbers. (Any value we put in for x in the original statement will give us a true inequality.)
(–∞, ∞) x

10. . x

11. Example: Solve the inequality. Graph and write in interval form.
– 7x + 9  – 5(x + 1)
– 7x + 9  – 5x – Use distributive property on right side.
5x – 7x + 9  – 5x + 5x – Add 5x from both sides.
– 2x + 9  – Simplify both sides.
– 2x + 9 – 9  – 5 – Subtract 9 from both sides.
– 2x  – Simplify both sides.
x < Divide both sides by – 2.
(–∞, 7] x 7

12. 3 x 5
Interval Notation (3, 7)

13. –7 + 3 < 2p – 3 + 3 ≤ 5 + 3 Add 3 to all three parts.
Example: Solve the inequality –7 < 2p – 3 ≤ 5. Graph the solution set and write it in interval notation.
–7 < 2p – 3 ≤ 5
–7 + 3 < 2p – ≤ Add 3 to all three parts.
–4 < 2p ≤ Simplify.
–2 < p ≤ Divide all three parts by 2.
Interval Notation (–2, 4] –2 p 4

15. Example
You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget.
I Introduction
Let x = represent the number of people
15x = cost for people
250 = set up fee
1200 = maximum you can spend

16. Example
II Body
Set up fee + (cost per person) × (number of people) ≤ 1200
(15) (x) ≤ 1200

17. Example III Conclusion
The number of people who can be invited must be 63 or less to stay within the budget.