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# Interval Notation Calculator + Online Solver With Free Steps

The **interval notation calculator** expresses the inequality based on the chosen topology and determines the distance between any two values.

The number line for the interval input is displayed by the** interval notation calculator**. Our online calculator for interval notation does calculations more quickly and displays the number line in a split second.

**What Is an Interval Notation Calculator?**

**The Interval Notation Calculator is an online tool that aids in displaying the given interval on a number line, shows the inequality by the chosen topology, and determines the distance between the two given integers.**

It is the method of writing subsets of the real number line, according to the mathematical definition. An example of interval notation includes the intervals expressed according to specified conditions.

For instance if we have the set $x |2 \leq x \leq 1$, it will be expressed as [2,1] by definition.

The formula for interval (set builder) notation is:

- n1 represents the first number
- n2 represents the second number

To solve the notation and find the interval values, use an online** interval notation solver**.

When a number is expressed as [a,x], it means that both “a” and “x” are part of a set. On the other hand, (a,x) denotes the omission of “a” and “x” from the collection.

The **half-closed symbol** “[b,y)” denotes that b is included but y is not. Similar to (b,y], which indicates that b is excluded and y is included in the collection, (b,y] will be recognized as half-open.

**How To Use an Interval Notation Calculator**

You can use the **Interval Notation Calculator** by following the given detailed guidelines, and the calculator will surely provide you with the desired results. You can therefore follow the given instructions to get the value of the variable for the given equation.

**Step 1**

Fill in the provided input boxes with the interval (closed or open interval).

**Step 2**

Click on the **“SUBMIT”** button to get the interval notation and also the whole step-by-step solution for the **Parametric to Cartesian Equation** will be displayed.

Finally, in the new window, the number line for the specified period will be displayed.

**How Does Interval Notation Calculator Work?**

The **I****nterval Notation Calculator** works by expressing the subset of real numbers using interval notation by the integers that bound them. Inequalities can be represented using this notation.

**Notations For Different Types of Intervals**

To represent the interval notation for various sorts of intervals, we can adhere to a set of rules and symbols. Let’s examine the various symbols that can be used to represent a specific kind of interval.

**Symbols Used for Interval Notation**

We use the following notations for various intervals:

- [ ]: When both endpoints are part of the set, this square bracket is used.
- ( ): When both endpoints are not included in the set, this round bracket is used.
- ( ]: When the right endpoint is included in the set but the left endpoint is excluded, a semi-open bracket is used.
- [ ): When the set’s left endpoint is included and its right endpoint is excluded, this semi-open bracket is likewise used.

**What Is Interval?**

The group of real numbers that lie between any two given real numbers is called **Interval** and is represented using interval notation. **Intervals** can be used to depict inequalities. Intervals can be divided into four categories.

If x and y are two endpoints and x y, the intervals can be classified into the following categories:

**Open Interval**

In this type of interval, the two ends are not included in this. The inequality is written as x < z < y if z is a number that falls between x and y. Round brackets are used to denote an **open interval**, i.e. (x, y).

**Closed Interval**

This type of interval includes both of the endpoints. As $x \leq z \leq y$, the inequality can be expressed. **Closed intervals** are expressed using square brackets, such as [x, y].

**Half Closed Right Interval**

Only the left endpoint is included in this kind of interval; the right endpoint is excluded. The inequality is x z y. The left side of the interval is enclosed in a square bracket, and the right side is enclosed in a round bracket, as in [x, y).

**Half Closed Left Interval**

The left endpoint is excluded and only the right endpoint is included while in this interval. In line with this, x < z ≤ y will be the inequality. The left side uses a round bracket and the right side will have a square bracket, i.e., (x, y].

The** Length of the interval** between the endpoints x and y can be calculated as follows:

**Length = y – x**

**Convert Inequality To Interval Notation**

To convert an** inequality to interval notation**, follow the steps shown below.

- Graph the interval’s solution set on a number line.
- The numbers should be written in interval notation with the smaller number on the left number line.
- Use the sign $-\infty$ if the set is unbounded on the left, and $\infty$ if it is unbounded on the right.

Let’s look at a few examples of inequality and convert them to interval notation.

- An Inequality $x \leq 3$ has interval notation $(-\infty, 3]$
- An Inequality $x < 5$ has interval notation $(-\infty, 5)$
- An Inequality $x \geq 2$ has interval notation $(2, \infty]$

**Represent Inequalities on a Number Line**

A **mathematical statement** known as an inequality compares two expressions using the concepts of greater than and less than. These statements employ unique symbols. Inequality should be read from left to right, much like the text on a page.

Large sets of solutions **are described by inequalities** in algebra. We have created some techniques to succinctly represent very big lists of numbers since there are occasionally an endless number of numbers that will fulfill an inequality.

You are presumably already aware of the **fundamental inequality** in a first way. For instance:

- The list of numbers less than 9 is shown by the expression $x \leq 9$.
- The symbol $-5 \leq t$ denotes all numbers greater than or equal to -5.

Keep in mind that whether you are searching for larger than or less than depends on whether the variable is placed to the left or right of the inequality sign.

**Important Notes on Interval Notation**

- The
**set of inequalities**is expressed using interval notation. - Open interval, closed interval, and half-open interval are the three different variants of
**interval notation**. - A bounded interval lacks the sign for
**infinity**. - An unbounded interval is the range that includes the infinity symbol.

**Solved Examples**

Let’s explore some examples to better understand the working of the **Interval Notation Calculator**.

**Example 1**

Check the solution to \[ x -10 \leq -12\]

### Solution

Substitute the endpoint -2 into the related equation as:

**x -10 $\leq$ -12**

**x -10 = -12**

Let’s check the following equality:

**-2 -10 = -12**

** -12 = -12**

Pick a value less than, such as, to check in the inequality given as:

** x -10 $\leq$ -12**

Let’s check the following inequality:

**-5 -10 $\leq$ -12**

**-15 $\leq$ -12**

It checks as:

**-5 -10 $\leq$ -12**

**x $\leq$ -2**

This is the solution to the following inequality:

**x -10 $\leq$ -12**

### Example 2

Find the domain of the following function:

\[f(x)=1/x^2 – 1\]

### Solution

The denominator being 0 is the only thing for which we need to be worried. We understand that x squared minus one cannot equal zero as a result. Because of this, x squared cannot equal one.

Then, x cannot be higher than or less than one if we take the square root of both sides. Therefore, we will be able to move from infinite to infinity when we specify our domain in interval notation. We’ll even go as far as the opposite.

\[ (- \infty, – 1) \cup (-1, 1) \cup (1, \infty) \]

As a result, this is our domain.

### Example 3:

What is the interval notation for the given function **f(x)=2** **by root over 3x+5?**

### Solution

In this equation, there is no negative radical, but there is a square root. We are aware that 3x +5 can never equal zero. It has to be more than zero or equal to it. It must be encouraging.

Additionally, as it is in a denominator, it cannot be zero or negative due to the radical in the expression. Therefore, when we solve this for “x” we observe that “3x” must be greater than -5.

In addition, we discover that “x” must be greater than $-\frac{5}{3}$ by dividing both sides by “3”. This means that you should start at -0.33 and work your way up to infinity in order to describe the domain using interval notation.

A parenthesis is always followed infinity. The only concern is whether we want to include the negative five-thirds, which we don’t.

\[(-\frac{5}{3}, \infty)\]

So, that gets a parenthesis as well, and there we have our domain.