1. Lesson 1 – 5
Solving Inequalities

2. Objectives Solve inequalities.
Solve real-world problems involving inequalities.

3. Trichotomy Property
For any two real numbers, a and b, exactly one of the following statements is true.

4. Addition Property of Inequality
For any real numbers a, b, and c:

5. Subtraction Property of Inequality
For any real numbers a, b, and c:

6. Solution Sets
These properties can be used to solve inequalities. The solution sets of inequalities in one variable can then be graphed on number lines.
Use a circle with an arrow to the left for < and an arrow to the right for >.
Use a dot with an arrow to the left for ≤ and an arrow to the right for ≥.

7. Example 1 {x | x > 9} Subtract 6x from each side.
Add 5 to each side.
We must write our answers in set notation.
{x | x > 9} 7 8 9

8. Multiplication Property of Inequality
For any real numbers a, b, and c:
If c is positive:
If c is negative:

9. Division Property of Inequality
For any real numbers a, b, and c:
If c is positive:
If c is negative:

10. Example 2 {y | y ≤ -8} Divide each side by -0.25.
Remember: When multiplying or dividing an inequality by a negative number, you must change the direction of the inequality.
{y | y ≤ -8}
-10 -9 -8

11. Interval Notation
The solution set of an inequality can also be described by using interval notation.
The infinity symbols are used to indicate that a set is unbounded in the positive or negative direction, respectively.
To indicate that an endpoint is not included is the set, parentheses are used.

12. Interval Notation Example 3
Write the solution set to the number line below using interval notation. 1 2

13. Interval Notation Example 4
Write the solution set to the number line below using interval notation. -2 -1

14. Example 5 Multiply each side by 9. Subtract m from each side.
Divide each side by -10. -1

15. Example 6 Distribute on each side. Simplify.
Subtract g from each side.
Subtract 8 from each side. 1 2

16. Example 7 Is this true? Cross Multiply. Distribute on each side.
Subtract 12x from each side.
Is this true?

17. “Or” Compound Inequalities
The graph of a compound inequality containing or is the union of the solution sets of the two inequalities.

18. Example 8 1 4

19. Example 9
Solve each inequality separately.
Now graph the union. -7 -1

20. “And” Compound Inequalities
A compound inequality containing the word and is true if and only if both inequalities are true.

21. Example 10 Graph x ≥ -1 and x < 2. x ≥ -1 x < 2 -1 ≤ x < 21 2
x < 2 -1 1 2
-1 ≤ x < 2 -1 1 2
The final graph is the intersection of the first two graphs.

22. Example 11 Subtract 7 from all “three” sides. Divide everything by 2.3 5
You can also rewrite this as 13 < 2x + 7 and 2x + 7 ≤ 17.