1. Algebra 1
Section 4.1

2. Inequalities
An inequality is a mathematical sentence stating that two quantities are not always equal.
There are five basic inequality symbols.

3. is greater than or equal to
Inequality Symbols
<
> ≠ ≤ ≥
is less than
is greater than
is not equal to
is less than or equal to
is greater than or equal to
4 < 6
5 > 3
-2 ≠ 4
-1 ≤ 8
6 ≥ 0

4. Trichotomy Property If a and b represent two real numbers,
then a = b, a > b, or a < b.

5. Inequalities
An inequality can be negated by putting a slash through it.
For example, x > 5 is read, “x is not greater than 5.”
This is equivalent to x ≤ 5.

6. Example 1 State an inequality statement equivalent to x ≤ y.
Answer: x > y.

7. Inequalities
Any value of the variable that makes an inequality true is a solution of the inequality.
There are typically an infinite number of solutions to each inequality.

8. Example 2 a + 3 > 6 2 + 3 > 6 is false. 3 + 3 > 6 is false.
4 + 3 > 6 is true.
4 is a solution.

9. Example 2 b. 6 ≥ 2b 6 ≥ 2(2) is true. 6 ≥ 2(3) is true.
6 ≥ 2(4) is false.
2 and 3 are solutions.

10. Example 2 1 – c < -2 1 – c ≥ -2 1 – 2 ≥ -2 is true.
1 – 4 ≥ -2 is false.
2 and 3 are solutions.

11. Graphing Inequalities
It is usually not possible to list all the solutions to an inequality. Therefore, we make a graph on a number line.

12. Graphing Inequalities
A circle is used when the number itself is not part of the solution set.
5 > x is usually rewritten: x < 5

13. Example 3
Graph x ≥ -3.
Since the inequality symbol includes equal to, a solid dot is placed at -3.

14. Example 4 Graph x ≠ 2. x > 2 or x < 2
Any number greater or less than 2 is a solution.

15. Example 6 Let c = price of an item c ≥ 9.99
(b) Let p = number of people
p ≤ 225

16. Homework: pp