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
Nonsolutions
Graph the solution set of x < 5:
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Representing Solutions to Inequalities
The solution to an inequality can be represented in four ways:
1) As an Inequality
Using the symbols >, <, ≤, ≥
Examples
𝑥>5
𝑥 < −4
𝑥≥7
𝑥 ≤ −2
x is greater than 5
x is less than -4
x is greater than or equal to 7
x is less than or equal to -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| 𝑥 < 5}
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) {𝑚|5 > 𝑚}
___________________________________________________
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Exercise
Write out in words the set described below.
1) {ℎ|ℎ ≤ −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)
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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 Notation
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Exercise
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
Graph
𝑎≥ 1
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Homework
Fill in the missing boxes in the chart below.
Set builder Notation
Set builder in
Words
Interval Notation
Graph
1) {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)
10)
The set of all b
such that b is less
than 8.
(−∞, −3)
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