Order Properties of the Real Numbers

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

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 >, ≥, ≤

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

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

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

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

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

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

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

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

Guided Practice Use interval notation to represent each graph below.