Day 1: Solution Sets of Equations and Inequalities
Warm-Up
Solve for x:
The Basics
An inequality is any statement that two quantities are not equal.
The quantities are compared using the following signs:
A solution to an inequality is any value that makes the inequality true. Often, an inequality has too many solutions to list individually, so we use a graph.
Example
List 3 solutions and 3 non-solutions to the inequality x < 5.
Solutions / NonsolutionsGraph the solution set of x < 5:
Representing Solutions to Inequalities
The solution to an inequality can be represented in four ways:
1) As an Inequality
Using the symbols >, <, ≤, ≥
Examples
x is greater than 5 / x>5x is less than -4 / x<-4
x is greater than or equal to 7 / x≥7
x is less than or equal to -2 / x≤-2
Note: If an inequality is inverted, we turn it around so that we can read it with the variable first.
5 > 3 is the same as 3 < 5.
3 > x is the same as x < 3.
2) Set-builder Notation
We can write the solution to an inequality as a set of all numbers that fit a certain description.
Inequality Set-builder notation
x < 5 x| x<5
This is read “the set of all x such that x is less than 5.”
Model Problem
Write out in words the set described below.
1) {x|x < 5} ______
2) {y|y ≥ 5} ______
3) m5> m} ______
Exercise
Write out in words the set described below.
1) hh≤ -6} ______
2) {r|r> 5} ______
3) {g|4 < g} ______
3) Using a Graph (Number Line)
Examples
1. {x|x=5} /2. {x|x=-6 or x=2} /
3. {x|x>-4} /
4. {x|x≠1} /
5. {x|-2≥x} /
6. {x|x ϵ R}
all real numbers
(infinitely many solutions) /
7. { }
empty set (no solutions) /
4) Interval Notation
An interval is a space between points, called endpoints. Interval notation represents a set of numbers using the endpoints and indicates whether the endpoints themselves are included in a set.
An open interval does not include the endpoints.
An open interval is indicated by parentheses: ()
A closed interval does include the endpoints.
A closed interval is indicated by square brackets: [ ]
An interval can also be half-open, including the endpoints on only one side.
When there is no endpoint or one or more sides of an interval, we use the symbols ∞ and – ∞.
(Note: these symbols always get parentheses on their side)
The symbol ∞means there is no highest number in the interval.
The symbol -∞ means there is no lowest number in the interval.
Examples
Graph / Set Notation / Interval NotationExercise
Write the inequality indicated by each graph in set-builder notation. Then write it in interval notation.
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Challenge! Sketch the graph of the solution to the inequality -2x < 6.
Give one number that is NOT in the solution set.
Exit Ticket
Express the given inequality in the ways indicated.
Inequality / Set-builder notation / Interval notation / Grapha≥ 1 /
Homework
Fill in the missing boxes in the chart below.
Set builder Notation / Set builder in Words / Interval Notation / Graph1) {d|d≥ 7}
2) {m|m=4}
3) / The set of all p such that m is not equal to -1.
4) / (-∞, 4]
5) {x|x> 1.5}
6) /
7) {d|5≥g}
8) /
9) / The set of all b such that b is less than 8.
10) / (-∞, -3)
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