Lesson Objectives

- Demonstrate an understanding of the number line
- Demonstrate an understanding of inequality symbols such as: <, ≤, >, and ≥
- Learn how to write the solution for an inequality using interval notation
- Learn how to graph an interval on the number line
- Learn how to write an interval using set-builder notation

## How to Write a Solution in Interval Notation or Set-Builder Notation

In our Algebra 1 course, we learned how to write a solution for an inequality using interval notation. We also learned how to graph an interval on the number line. In this lesson, we will review these concepts and also learn how to write a solution for an inequality using set-builder notation.

Let's first look at how we deal with a typical linear equation in one variable.

2x - 13 = 1

2x = 14

x = 7

We can notate this solution in a few different ways. For most of algebra, stating that x = 7 is generally as far as we need to go. We can also write our answer in solution-set notation. The solution set is just a set with the solution or solutions as its elements. In this case, our solution set would be written as: {7}

Additionally, we could use set-builder notation. We will cover this in more detail further in the lesson. For now, this would be written as:

{x | x = 7}

Which is read "the set of all x such that x is 7".

Lastly, we could show this solution graphically. We would draw a filled-in circle at the number 7 on the number line: All of these ways to show our solution of x = 7 are giving us the same information. In order for our equation to be true, x must be 7. If x is any other value, the equation will not be satisfied. When we work with inequalities, there is generally not one single solution. The solution to an inequality is normally a range of values. Let's suppose we had a very simple inequality such as:

x > 7

What values could be plugged in for x?

Essentially, anything that is larger than 7. This means there is an infinite number of solutions. x could be replaced with 7.00001, 8, 127, or a really large number such as 138,512,333. It really doesn't matter, so long as the value is larger than 7. In order to show this type of solution set, we generally use graphing, interval notation, or set-builder notation. Let's start out by thinking about how to graph an interval on the number line.

x > 7

Boundary:

x = 7

The boundary separates the solution region from the non-solution region. On one side of the boundary, all numbers will satisfy the inequality and on the other side, all numbers will not satisfy the inequality. At the boundary, we have a few different scenarios. If we are dealing with a strict inequality:

< or >

We use a parenthesis that faces toward the solution region or an open circle

We use these symbols to indicate that the boundary is not part of the solution. On the other hand, we may work with non-strict inequalities. When we are dealing with non-strict inequalities:

≤ or ≥

We use a bracket that faces toward the solution region or a closed circle

We use these symbols to indicate that the boundary is part of the solution. This is because of the "or equal to" part of the non-strict inequality. If we want to graph:

x > 7

We would place a parenthesis facing to the right at 7 and then shade all values to the right of 7. We will also shade the right arrow to indicate that all numbers are solutions out to positive infinity: Additionally, we can show this using an open circle: If we changed our problem up to use a non-strict inequality:

x ≥ 7

We would change to either a bracket or a filled-in circle:

(a , » a left parenthesis "(" indicates the value to its right is excluded. This means our smallest value in the interval is any value that is larger.

, b) » a right parenthesis ")" indicates the value to its left is excluded. This means our largest value in the interval is any value that is smaller.

1 < x < 5

To write this in interval notation, first think about the smallest value that x can take on. Since x is larger than 1, we start by placing a left parenthesis followed by a 1 and then a comma:

(1,

Now we think about the largest value. Since x is less than 5, we place a 5 followed by a right parenthesis:

(1, 5)

This interval (1,5) tells us that x can be any value larger than 1 up to but not including the number 5. The parenthesis in each case excludes the numbers. Essentially, x is between 1 and 5. If we change this interval up to use non-strict inequalities:

[a , » a left bracket "[" indicates the value to its right is included. This means our smallest value in the interval is the value to the right of the left bracket.

, b] » a right bracket "]" indicates the value to its left is included. This means our largest value in the interval is the value to the left of the right bracket.

1 ≤ x ≤ 5

[1,5]

This interval [1,5] tells us that x can be any value that is between 1 and 5, with 1 and 5 being included. The bracket in each case is what includes the number. Infinity "∞" is a concept in math which deals with a quantity that is not countable. When we see something such as:

x > 3

x can be any number larger than 3. Since there is no largest value, we will use infinity in its place. Infinity is always placed next to a parenthesis.

(3, ∞)

Additionally, we will have to use negative infinity "- ∞". Let's suppose we had something such as:

x < 3

(-∞, 3)

Here we used negative infinity "- ∞" in the place of the smallest number since there isn't one. x can take on any value that is smaller than 3.

{x | x has some property}

The first part is read as: "the set of all x"

"|" is read as: "such that"

The last part is read as: "x has some property"

For right now, when we hear "the set of all x", we are saying x is some real number. We then become more specific by saying "such that" and lastly we give our condition. So x is a real number such that it has a specific property.

Let's say we had the problem:

x > 14

We can write this in set-builder notation as:

{x | x > 14}

Basically, this is just saying that x is any real number that is larger than 14. Let's look at a few examples that deal with interval notation, graphing intervals, and set-builder notation.

Example 1: Write each in interval notation, set-builder notation, and graph the interval

x > 9

Interval Notation:

(9, ∞)

Set-Builder Notation:

{x | x > 9}

Graphing the Interval: Example 2: Write each in interval notation, set-builder notation, and graph the interval

-4 ≤ x < 12

Interval Notation:

[-4, 12)

Set-Builder Notation:

{x | -4 ≤ x < 12}

Graphing the Interval: Example 3: Write each in interval notation, set-builder notation, and graph the interval

-19 ≤ x ≤ -1

Interval Notation:

[-19, -1]

Set-Builder Notation:

{x | -19 ≤ x ≤ -1}

Graphing the Interval:

Let's first look at how we deal with a typical linear equation in one variable.

2x - 13 = 1

2x = 14

x = 7

We can notate this solution in a few different ways. For most of algebra, stating that x = 7 is generally as far as we need to go. We can also write our answer in solution-set notation. The solution set is just a set with the solution or solutions as its elements. In this case, our solution set would be written as: {7}

Additionally, we could use set-builder notation. We will cover this in more detail further in the lesson. For now, this would be written as:

{x | x = 7}

Which is read "the set of all x such that x is 7".

Lastly, we could show this solution graphically. We would draw a filled-in circle at the number 7 on the number line: All of these ways to show our solution of x = 7 are giving us the same information. In order for our equation to be true, x must be 7. If x is any other value, the equation will not be satisfied. When we work with inequalities, there is generally not one single solution. The solution to an inequality is normally a range of values. Let's suppose we had a very simple inequality such as:

x > 7

What values could be plugged in for x?

Essentially, anything that is larger than 7. This means there is an infinite number of solutions. x could be replaced with 7.00001, 8, 127, or a really large number such as 138,512,333. It really doesn't matter, so long as the value is larger than 7. In order to show this type of solution set, we generally use graphing, interval notation, or set-builder notation. Let's start out by thinking about how to graph an interval on the number line.

### Graphing an Interval on the Number Line

When graphing an interval on the number line, we first must understand about boundaries. When we work with inequalities, the boundary can be found by replacing the inequality symbol with an equality symbol. If we use our earlier example of x > 7, to find our boundary we would replace the greater than with an equality:x > 7

Boundary:

x = 7

The boundary separates the solution region from the non-solution region. On one side of the boundary, all numbers will satisfy the inequality and on the other side, all numbers will not satisfy the inequality. At the boundary, we have a few different scenarios. If we are dealing with a strict inequality:

< or >

We use a parenthesis that faces toward the solution region or an open circle

We use these symbols to indicate that the boundary is not part of the solution. On the other hand, we may work with non-strict inequalities. When we are dealing with non-strict inequalities:

≤ or ≥

We use a bracket that faces toward the solution region or a closed circle

We use these symbols to indicate that the boundary is part of the solution. This is because of the "or equal to" part of the non-strict inequality. If we want to graph:

x > 7

We would place a parenthesis facing to the right at 7 and then shade all values to the right of 7. We will also shade the right arrow to indicate that all numbers are solutions out to positive infinity: Additionally, we can show this using an open circle: If we changed our problem up to use a non-strict inequality:

x ≥ 7

We would change to either a bracket or a filled-in circle:

### Writing a solution in Interval Notation

Interval notation is a convenient way to show an interval on a number line. We will use: "(" or ")" to exclude a number from the interval and "[" or "]" to include a number in the interval.(a , » a left parenthesis "(" indicates the value to its right is excluded. This means our smallest value in the interval is any value that is larger.

, b) » a right parenthesis ")" indicates the value to its left is excluded. This means our largest value in the interval is any value that is smaller.

1 < x < 5

To write this in interval notation, first think about the smallest value that x can take on. Since x is larger than 1, we start by placing a left parenthesis followed by a 1 and then a comma:

(1,

Now we think about the largest value. Since x is less than 5, we place a 5 followed by a right parenthesis:

(1, 5)

This interval (1,5) tells us that x can be any value larger than 1 up to but not including the number 5. The parenthesis in each case excludes the numbers. Essentially, x is between 1 and 5. If we change this interval up to use non-strict inequalities:

[a , » a left bracket "[" indicates the value to its right is included. This means our smallest value in the interval is the value to the right of the left bracket.

, b] » a right bracket "]" indicates the value to its left is included. This means our largest value in the interval is the value to the left of the right bracket.

1 ≤ x ≤ 5

[1,5]

This interval [1,5] tells us that x can be any value that is between 1 and 5, with 1 and 5 being included. The bracket in each case is what includes the number. Infinity "∞" is a concept in math which deals with a quantity that is not countable. When we see something such as:

x > 3

x can be any number larger than 3. Since there is no largest value, we will use infinity in its place. Infinity is always placed next to a parenthesis.

(3, ∞)

Additionally, we will have to use negative infinity "- ∞". Let's suppose we had something such as:

x < 3

(-∞, 3)

Here we used negative infinity "- ∞" in the place of the smallest number since there isn't one. x can take on any value that is smaller than 3.

### Set-Builder Notation

Lastly, we want to look at set-builder notation. This type of notation comes up pretty often in college algebra and higher-level math. Essentially set-builder notation looks like:{x | x has some property}

The first part is read as: "the set of all x"

"|" is read as: "such that"

The last part is read as: "x has some property"

For right now, when we hear "the set of all x", we are saying x is some real number. We then become more specific by saying "such that" and lastly we give our condition. So x is a real number such that it has a specific property.

Let's say we had the problem:

x > 14

We can write this in set-builder notation as:

{x | x > 14}

Basically, this is just saying that x is any real number that is larger than 14. Let's look at a few examples that deal with interval notation, graphing intervals, and set-builder notation.

Example 1: Write each in interval notation, set-builder notation, and graph the interval

x > 9

Interval Notation:

(9, ∞)

Set-Builder Notation:

{x | x > 9}

Graphing the Interval: Example 2: Write each in interval notation, set-builder notation, and graph the interval

-4 ≤ x < 12

Interval Notation:

[-4, 12)

Set-Builder Notation:

{x | -4 ≤ x < 12}

Graphing the Interval: Example 3: Write each in interval notation, set-builder notation, and graph the interval

-19 ≤ x ≤ -1

Interval Notation:

[-19, -1]

Set-Builder Notation:

{x | -19 ≤ x ≤ -1}

Graphing the Interval:

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