# Inequality to Interval Notation Calculator

Created by Anna Szczepanek, PhD
Reviewed by Wojciech Sas, PhD and Jack Bowater
Last updated: Jun 05, 2023

This inequality to interval notation calculator is the first place you should visit when you need to convert between these two very popular types of mathematical notation.

Importantly, Omni's inequality to interval notation converter can work both ways! That is, it can teach you how to write an inequality in interval notation, and also how to convert interval notation to inequality notation. It can even deal with compound inequalities!

Once you're done here, take one more step in your mathematical journey and discover how to graph inequalities on a number line!

## What is the interval notation in math?

An interval is a subset of real numbers that consists of all numbers contained between two given numbers called the endpoints of the interval.

Intervals are directly linked to inequalities: the numbers contained in an interval are exactly those that satisfy certain inequalities related to the endpoints of our interval. For example, the set of numbers x satisfying the inequality 0 < x < 7 is the set that contains all numbers that are simultaneously greater than 0 and less than 7, so the interval with the endpoints 0 and 7.

💡 Intervals may look innocent and very simple, but they are vital in various branches of science! Discover one of their most important applications in statistics with Omni's confidence interval calculator!

## What are the types of intervals?

There are different types of intervals, depending on whether or not each of the endpoints belongs to the interval. In general, there are three types of intervals:

• Open intervals - Do not include the endpoints;
• Closed intervals - Do include the endpoints; and
• Half-open intervals - Include exactly one of the endpoints.

To determine whether or not a given endpoint is included, just take a look at the symbols used to denote the interval - parentheses (round brackets) mean the endpoint is not included, while square brackets mean the endpoint is included:

• Open interval notation: (a,b);
• Closed interval notation: [a,b]; and
• Half-open interval notation: (a,b] or [a,b).

## How to use this inequality to interval notation calculator?

Here we explain how our inequality to interval notation calculator works:

1. Start by choosing the calculator mode, that is, the conversion direction:

• From interval notation to inequality; or
• From inequality to interval notation
2. For the interval to inequality mode, pick the interval type and enter the endpoints in the appropriate fields of the calculator. The result - the inequality corresponding to your interval - will appear underneath.

3. For the inequality to interval notation converter, first choose the inequality type:

• One-sided;
• Two-sided; or
• Compound,

and then choose the exact form of the inequality you wish to convert to interval notation.

4. The last bit of information that our inequality to interval notation calculator requires to work properly is the value(s) of endpoint(s): enter it/them into the corresponding field(s). Again, the result - this time the intervals corresponding to your inequality - will appear at the bottom of our tool.

If possible, the calculator will simplify the solution and display the subset of real numbers corresponding to your inequalities in the most compact way possible!

As you can see, it's really simple to write the solution to a given inequality in interval notation with the help of our inequality to interval notation converter!

However, life's not always that easy, and sometimes (on a math test most probably) you won't have access to the Internet! 😱 It's better to be prepared, so let's now discuss how to put an inequality into interval notation by hand and how to convert interval notation to inequality notation.

## How to convert inequality notation to interval notation?

You can very quickly write an inequality in interval notation with the help of the following table, which you can think of as an interval-inequality dictionary 😉

Interval

Inequality

(a, b)

a < x < b

[a, b]

a ≤ x ≤ b

(a, b]

a < x ≤ b

[a, b)

a ≤ x < b

[a, ∞)

x ≥ a

(a, ∞)

x > a

(-∞, a)

x < a

(-∞, a]

x ≤ a

The first four interval types above are bounded (they have two endpoints a and b), the other four are unbounded (they have the starting point a and extend to plus infinity or minus infinity).

It's easy to write the solution to the given inequality in interval notation, isn't it? However, this task becomes a bit more challenging if there are more inequalities. In the next section, we'll learn how to convert compound inequalities to interval notation.

## How to solve compound inequalities in interval notation?

By a compound inequality we mean two inequalities that are joined by and/or. To convert a compound inequality to interval notation, follow these steps:

1. Rewrite the two inequalities as a single inequality, taking into account the conjunction between them:
• And means that both inequalities must be satisfied; and
• Or means that at least one of the equalities must be satisfied.
2. As a result you get a one- or two-sided inequality. You now need to convert this inequality to interval notation.
3. That's it! You may verify your calculation using Omni's inequality to interval notation calculator.

As you can see, it's not hard to convert a compound inequality to interval notation. The most tricky part is Step 1. (rewriting the two inequalities into a single one). If you need a bit more help with that - keep reading! In what follows we explain in more detail how to rewrite the compound inequality as a single inequality and then how to express it in interval notation.

## Compound inequalities with And

Here we discuss how to do interval notation for inequalities with conjunction And. Recall that And means that x has to satisfy both inequalities in order to be part of the final result.

We will consider two subcases, corresponding to whether the inequalities in question have the same direction (e.g., x > a and x ≥ b) or the opposite directions (e.g., x < a and x ≥ b).

#### Same direction

If the inequalities are of the same type, that is, both are strict (like >) or both are not strict (like ≥), then the resulting inequality is also of this type. As a consequence, we will get an unbounded interval as our final result. To determine its endpoint, you need to find the more restrictive condition. In the case of > and ≥ inequalities, it is bigger numbers that are more restrictive, so the endpoint is the maximum of the two initial endpoints. In the case of < and ≤ inequalities, smaller numbers are more restrictive, so the endpoint is the minimum of the initial endpoints. We summarize this in the below table:

 x ≥ a and x ≥ b = [max(a,b), ∞) x > a and x > b = (max(a,b), ∞) x < a and x < b = (-∞, min(a,b)) x ≤ a and x ≤ b = (-∞, min(a,b)]

If the inequalities are of opposite types, then the situation becomes slightly more complicated, because the type (strict or not strict) of the resulting inequality now depends on the type of the more restrictive inequality.

Assuming a ≥ b

x ≥ a

and

x > b

=

[a, ∞)

x > a

and

x ≥ b

=

(a, ∞)

x ≤ a

and

x < b

=

(-∞, b)

x < a

and

x ≤ b

=

(-∞, b]

Assuming a < b

x ≥ a

and

x > b

=

(b, ∞)

x > a

and

x ≥ b

=

[b, ∞)

x ≤ a

and

x < b

=

(-∞, a]

x < a

and

x ≤ b

=

(-∞, a)

#### Opposite directions

Here, the final result will be either a bounded interval or an empty set. In the former case, looking at the type of inequalities, we can determine whether the final interval is open, closed, or half-open. Here are the details:

Assuming a < b

x ≥ a

and

x ≤ b

=

[a, b]

x ≥ a

and

x < b

=

[a, b)

x > a

and

x ≤ b

=

(a, b]

x > a

and

x < b

=

(a, b)

Assuming a = b

x ≥ a

and

x ≤ b

=

{a}

x ≥ a

and

x < b

=

x > a

and

x ≤ b

=

x > a

and

x < b

=

Finally, if a > b, then we always get the empty set ∅.

## Compound inequalities with Or

In this final section, we discuss how to do interval notation for inequalities with the conjunction Or. Recall that Or means that x has to satisfy either of the two inequalities in order to be a part of the final result.

Again, we consider two subcases: same direction inequalities and opposite directions inequalities.

#### Same direction

As in the And case, if the inequalities are of the same type, then the resulting inequality is also of this type, so we obtain an unbounded interval as our final result. Its endpoint is determined by the less restrictive of the conditions in question. In the case of > and ≥ inequalities, smaller numbers are less restrictive, so the endpoint is the minimum of the initial endpoints. When we have < and ≤ inequalities, then bigger numbers are less restrictive, so the endpoint is the maximum of the initial endpoints:

 x ≥ a or x ≥ b = [min(a,b), ∞) x > a or x > b = (min(a,b), ∞) x < a or x < b = (-∞, max(a,b)) x ≤ a or x ≤ b = (-∞, max(a,b)]

If the inequalities are of the opposite types, then the type of the less restrictive inequality determines the type of the resulting inequality, and therefore that of the final interval:

Assuming a ≥ b

x ≥ a

or

x > b

=

(b, ∞)

x > a

or

x ≥ b

=

[b, ∞)

x ≤ a

or

x < b

=

(-∞, a]

x < a

or

x ≤ b

=

(-∞, a)

Assuming a < b

x ≥ a

or

x > b

=

[a, ∞)

x > a

or

x ≥ b

=

(a, ∞)

x ≤ a

or

x < b

=

(-∞, b)

x < a

or

x ≤ b

=

(-∞, b]

#### Opposite directions

Here, the final result will be either the union of two disjoint unbounded intervals or the full interval (-∞,∞), so, in other words, the set of all real numbers ℝ:

Assuming a > b

x ≥ a

or

x ≤ b

=

(-∞, b] ∪ [a ,∞)

x ≥ a

or

x < b

=

(-∞, b) ∪ [a ,∞)

x > a

or

x ≤ b

=

(-∞, b] ∪ (a ,∞)

x > a

or

x < b

=

(-∞, b) ∪ (a ,∞)

Assuming a = b

x ≥ a

or

x ≤ b

=

(-∞,∞)

x ≥ a

or

x < b

=

(-∞,∞)

x > a

or

x ≤ b

=

(-∞,∞)

x > a

or

x < b

=

(-∞,∞)\{a}

In case you're not familiar with the notation (-∞,∞)\{a}, it means "all numbers except a".

Finally, if a < b, then in all four cases we get the full interval (-∞,∞).

And that's it when it comes to learning how to solve compound inequalities in interval notation! To verify if you've got the concept well, use our inequality to interval notation calculator to generate some examples!

Anna Szczepanek, PhD
Conversion type
Inequality to interval
Choose the inequality type
One-sided
x < a
a
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