1. Order Properties of the Real Numbers

2. Inequality Notation *DEFINITION
Suppose that 𝑎 and 𝑏 are any real numbers
We will say that “𝑎 is to the left of 𝑏” on the number line means the same as “𝑎<𝑏” or “𝑎 is less than 𝑏”
We will say that “𝑏 is to the right of 𝑎” on the number line means the same as “𝑏>𝑎” or “𝑏 is greater than 𝑎”
Note that the symbols < and > are arrowheads; the first points to the left, the second points to the right

3. The Order Properties
*Suppose that 𝑎, 𝑏, and 𝑐 are any real numbers. Then:
Exactly ONE of the following must be true: 𝑎<𝑏, 𝑎=𝑏, 𝑎>𝑏
If 𝑎<𝑏 and 𝑏<𝑐, then 𝑎<𝑐
If 𝑎<𝑏, then 𝑎+𝑐<𝑏+𝑐
If 𝑎<𝑏 and 𝑐>0, then 𝑎𝑐<𝑏𝑐 (𝑐 is positive)
If 𝑎<𝑏 and 𝑐<0, then 𝑎𝑐>𝑏𝑐 (𝑐 is negative)
The properties are the same for >, ≥, ≤

4. Graphing Inequalities
To graph an inequality on the number line, shade that part of the number line corresponding to all the numbers that make the inequality true
*If the inequality is > or <, then use an open circle, ○, at the start point
*If the inequality is ≥ or ≤, then use a closed circle, ●, at the start point
*Graph the inequality 𝑥<5
*Graph the inequality 𝑥≥−2

5. Compound Inequalities
A compound inequality combines two (or more) inequalities using the words “and” or “or”
*Compound inequalities that use “and” are called conjunctions and include all the numbers that make both inequalities true
*Compound inequalities that use “or” are called disjunctions and include all the numbers that make either of the inequalities true
*Graph the inequality 𝑥>2 AND 𝑥<5
*Graph the inequality 𝑥<2 OR 𝑥>5

6. Guided Practice Graph the following inequalities. 𝑥≤0 𝑥≤4 OR 𝑥>8
−1≤𝑥≤1
−2<𝑦≤3

7. Using Set-Builder Notation
Recall that set-builder notation has the form 𝑥 "something about 𝑥"
We can express the solution set of an inequality in set-builder notation
Write the following inequalities in set-builder notation:
𝑥>5
−5≤𝑦≤5

8. Guided Practice
Translate the following phrases to set-builder notation.
The set of all real numbers 𝑦 such that 𝑦 is less than −1
The set of all real numbers 𝑥 such that 𝑥 is greater than 3 and 𝑥 is less than 10
The set of all real numbers 𝑧 such that 𝑧 is less than zero or 𝑧 is greater than 1

9. Guided Practice Translate the following sets into a sentence. {𝑛|𝑛≤−3}
{𝑝|𝑝<1 𝑜𝑟 𝑝≥5}
{𝑥|0≤ 𝑥≤10}

10. Interval Notation
Another way to represent an inequality is by interval notation
Interval notation uses parentheses and/or brackets to represent a set of numbers either between two other numbers, or all numbers to the left or right of a number
*Parentheses ( ) correspond to < and >; brackets [ ] correspond to ≤ and ≥
*You will also use the symbol ∞ to indicate that the numbers continue indefinitely to the right, and the symbol −∞ to indicate that the numbers continue indefinitely to the left

11. Interval Notation Examples
𝑥>5 is the same as 5,∞ ; note that the parenthesis at the left indicates that 5 is not part of the set
𝑥≤0 is the same as (−∞,0]; the bracket on the right indicates that zero is in the set
−6<𝑥≤1 is the same as (−6,1]
𝑥<2 𝑜𝑟 𝑥>3 is the same as (−∞,2) 𝑜𝑟 (3,∞); the word or can also be represented by the symbol ∪, so the above can be written as −∞,2 ∪(3,∞)
The symbol ∪ is call the union and is used for disjunctions

12. Guided Practice
Rewrite the following inequalities using interval notation.
𝑛≤8
4≤𝑦≤7
𝑥<2 𝑜𝑟 𝑥>10
𝑏>−1

13. Guided Practice
Use interval notation to represent each graph below.