www.VirtualNerd.com

How Do You Write Inequalities in Set Builder Notation?

Convert these two solutions into set-builder notation:
1st Solution: "t is the set of all numbers, such that t is greater than 80"
2nd Solution: "y is the set of all numbers, such that y is greater than 24 and less than or equal to 53"

Summary

  1. Set-builder notation has three parts
  2. In math, anything contained within the curly braces is known as a set
  3. The vertical line '|' represents the words "such that"
  4. The inequality symbol tells us we are dealing with an inequality
  5. The '<' symbol means LESS THAN
  6. The '>' symbol means GREATER THAN
  7. The '≤' symbol means LESS THAN OR EQUAL TO
  8. Because the 2nd solution is a compound inequality, we can write the set-builder notation more than one way

Notes

    1. There are three necessary parts to remember for set-builder notation
    1. There are three necessary parts to remember for set-builder notation
    2. Enclosing the set in curly braces is one part to remember
    1. There are three necessary parts to remember for set-builder notation
    2. Enclosing the set in curly braces is one part to remember
    3. Curly braces look like '{ }'
    1. There are three necessary parts to remember for set-builder notation
    2. Enclosing the set in curly braces is one part to remember
    3. Curly braces look like '{ }'
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    4. The vertical line '|' means "such that"
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    4. The vertical line '|' means "such that", and is what actually "defines" the variable 'x' in this case
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    4. The vertical line '|' means "such that", and is what actually "defines" the variable 'x' in this case
    1. There are three necessary parts to remember for set-builder notation
    2. Identifying the variable that represents the set of numbers is a necessary part of set-builder notation
    3. 'x' is our variable here
    4. The vertical line '|' means "such that", and is what actually "defines" the variable 'x' in this case
    5. The specifics of what makes the set unique come after the vertical line
    1. There are three necessary parts to remember for set-builder notation
    2. The inequality that defines the numbers contained in the set 'x' is a necessary part to remember
    3. 'x' is our variable representing the set's numbers
    1. There are three necessary parts to remember for set-builder notation
    2. The inequality that defines the numbers contained in the set 'x' is a necessary part to remember
    3. 'x' is our variable representing the set's numbers
    4. 'y' is also a variable here
    5. The inequality defining the set is 'x<y'
    6. The '<' symbol means LESS THAN
    1. There are three necessary parts to remember for set-builder notation
    2. The inequality that defines the numbers contained in the set 'x' is a necessary part to remember
    3. 'x' is our variable representing the set's numbers
    4. 'y' is also a variable here
    5. The inequality defining the set is 'x<y'
    6. The '<' symbol means LESS THAN
    7. Putting it all together, we get '{x | x<y}'
    1. There are three necessary parts to remember for set-builder notation
    2. The inequality that defines the numbers contained in the set 'x' is a necessary part to remember
    3. 'x' is our variable representing the set's numbers
    4. 'y' is also a variable here
    5. The inequality defining the set is 'x<y'
    6. The '<' symbol means LESS THAN
    7. Putting it all together, we get '{x | x<y}'
    1. The variable represents all the numbers included in the set
    2. The vertical line '|' says we're going to define the variable
    3. The inequality is what actually defines what numbers will be included in the set
    1. The variable represents all the numbers included in the set
    2. The vertical line '|' says we're going to define the variable
    3. The inequality is what actually defines what numbers will be included in the set
    4. The variable, vertical line, and inequality are all contained in the curly braces, since we're defining a set
    1. Let's look at two example solutions
    1. Our variable representing the set of numbers is 't' in this case
    2. The phrase "t is GREATER THAN 80" can be rewritten as 't>80'
    1. Remember that curly braces are used when dealing with sets
    2. Our variable representing the set of numbers is 't' in this case
    1. The vertical line '|' says we're going to define what numbers 't' represents in the set
    1. The inequality is what actually defines what numbers will be included in the set
    2. Our inequality is 't>80', meaning that this set will include all numbers greater than 80
    1. The notation is: '{t | t>80}'
    1. The phrase "y is GREATER THAN 24 and LESS THAN OR EQUAL TO 53" can be rewritten as 'y>24 AND y≤53'
    2. Our variable representing the set of numbers is 'y' in this case
    1. Remember that curly braces are used when dealing with sets
    2. Our variable representing the set of numbers is 'y' in this case
    3. The vertical line '|' says we're going to define what numbers 'y' represents in the set
    1. We have a compound inequality, so we have two inequalities to deal with in this example
    1. Our variable representing the set of numbers is 'y' in this case
    2. The '>' symbol means GREATER THAN
    1. So 'y>24' goes after the '{y |' part in the set-builder notation
    1. We use 'AND' to link the two inequalities together and form a compound inequality
    2. So 'AND' goes after the '{y | y>24' part in the set-builder notation
    1. Our variable representing the set of numbers is 'y' in this case
    2. The '≤' symbol means LESS THAN OR EQUAL TO
    1. Don't forget to put a curly brace on the end to close the set!
    2. Remember that curly braces are used when dealing with sets
    1. The notation is: '{y | y>24 AND y≤53}'
    1. We can simplify the notation slightly by writing the notation in a different way
    1. We can simplify the notation slightly by writing the notation in a different way
    1. Remember that curly braces are used when dealing with sets
    1. The order of terms in an inequality can be flipped around as long as you remember to flip the inequality symbol, too!
    1. The order of terms in an inequality can be flipped around as long as you remember to flip the inequality symbol, too!
    1. The order of terms in an inequality can be flipped around as long as you remember to flip the inequality symbol, too!
    2. Here, we're looking at the first half of our compound inequality, 'y>24'
    3. So 'y>24' can also be written as '24<y'
    1. The order of terms in an inequality can be flipped around as long as you remember to flip the inequality symbol, too!
    2. Here, we're looking at the first half of our compound inequality, 'y>24'
    3. So 'y>24' can also be written as '24<y'
    1. Our second inequality is 'y≤53'
    1. We can combine the two inequalities by dropping the 'AND' and dropping a 'y'
    2. The '≤' symbol means LESS THAN OR EQUAL TO
    3. So this will give reduce our compound inequality to: '24<y≤53'
    1. Don't forget to put a curly brace on the end to close the set!
    1. The notation is: '{y | 24<y≤53}'
    1. We went from '{y | y>24 AND y≤53}' to '{y | 24<y≤53}'
    1. We went from '{y | y>24 AND y≤53}' to '{y | 24<y≤53}'
    1. Remember that curly braces are used when dealing with sets
    2. We use variables to represent numbers in the set