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# Inequality Calculator + Online Solver With Free Steps

The **Inequality Calculator **is a tool used to calculate the interval of the unknown variable in a linear inequality.

The **calculator** takes the mathematical expression for the inequality as input and in return, it finds the interval notation and number line representation with an inequalities plot.

## What Is the Inequality Calculator?

**The Inequality Calculator is an online calculator that allows you to determine the intervals for linear inequality problems.**

**Linear Inequality** is an expression that uses symbols of inequality to perform a comparison between two algebraic terms. It is easy to solve these inequalities manually but for this, you need to use basic mathematical techniques and do some calculations.

Therefore we offer you this advanced **Inequality Calculator **that can solve any kind of linear equality within a few seconds. You only need to enter the inequality; there is no need to perform any maths.

Mathematicians and students can deal with linear equality problems without any hassle using this **powerful** tool. Unlike other modern tools, you don’t need to buy a subscription for using it.

This **calculator** is entirely free and can be accessed 24/7 with any appropriate browser. It is an effective and reliable tool because it provides the **perfect** solutions for your problem.

We are confronted with **linear inequalities** almost every day. It is mainly used in finding ranges of a parameter like maximum transaction from debit card, area of a field, calculating speed limits, persons in a lift, etc.

To know further information about the procedure and working mechanism of the calculator you may refer to the next sections.

## How To Use the Linear Inequality Calculator?

To Use the **Inequality Calculator, **we plug in the expression of inequality required by the calculator.

The calculator’s front end consists of an empty box for the **input** and a click-button for acquiring the **solution**. This tool is simple enough for anyone to use. It can handle only one linear inequality at a time.

You must follow the given detailed guidelines, and the calculator will surely provide you with the desired results.

### Step 1

Enter the linear equality in the given space. Make sure to use the correct signs of inequality according to your problem.

### Step 2

After entering the expression, press the “**Submit”** button to start the computation.

### Output

The calculator gives the solution to the problem in several steps. In the first step, it gives the input information where the user can once again validate the input.

Then the **inequality plot** is shown. Here, the two sides of an inequality are considered as separate terms and their respective graphs are plotted.

It gives the **solution **to the inequality and the proper **notation** of the interval for the unknown variable. Also, it provides the various alternate forms of the obtained interval.

In addition to these solutions, the calculator has an additional feature of **number line** representation that allows users to visualize the obtained interval in a single plane of the variable.

## How Does the Inequality Calculator Work?

The inequality calculator works by solving the** linear inequalities** and finding its solution for the required variables. It also provides the inequality graph and its solution on the number line.

The appropriate use of this inequality calculator can be made possible when there is knowledge about inequality and its types.

### What Is an Inequality?

Inequalities are mathematical expressions that are **not equal** on both sides. It is the relationship of expression which have a non-equal comparison.

The equal sign in between the equation is replaced by greater than, greater than or equal to, less than, less than, or equal to sign.

There are different types of inequalities such as Polynomial inequalities, Absolute value inequalities, and Rational inequalities.

#### Polynomial Inequalities

Polynomial inequalities contain** polynomials **on both sides of the inequality. Polynomial inequalities are further divided into different types but the most important ones are Linear inequalities and Quadratic inequalities.

This calculator focuses on solving** linear** inequalities therefore the explanation and method of solving linear inequalities are given below.

#### Linear Inequalities

The algebraic inequality in which two** linear polynomials** are compared using the inequality symbols is known as** linear inequality**. The expression on both sides of inequality must be a polynomial having the highest power equal to one.

### Rules of Inequalities

The four basic arithmetic operators are applied to linear inequalities for solving them. However, there are some rules for these operators that should know before using them.

#### Addition Rule

The addition rule states that when a number is added on both sides of inequality there is **no change** in the inequality symbol. For instance, adding a number in the inequality ‘x < y’ results in ‘x+a < y+a’.

#### Subtraction Rule

When a constant is subtracted from the inequality, the inequality sign **does not** change according to the subtraction rule. If there is inequality such as ‘z > x’ then after subtracting a number it gives ‘z-b > x-b’.

#### Multiplication Rule

The multiplication rule changes the inequality symbol according to the positive or negative number that is multiplied. If the **positive** number is multiplied on both sides of an inequality, the symbol** does no**t change.

Whereas, multiplication with a** negative** number results in a** change** of the inequality symbol. For example, the inequality ‘y > z’ when multiplied by the negative constant ‘a < 0’ gives ‘y*a < z*a’.

#### Division Rule

The division rule implies that the inequality symbol **does not change** when there is a division of** positive **numbers. However, when a** negative **number is divided into both sides of inequality, then the symbol is **reversed**.

If the inequality ‘x < y’ is divided by a negative constant ‘c < 0’ then it results in ‘(x/c) > (y/c)’.

### Solving the Linear Inequality

The **linear inequalities** can be solved by simplifying the inequalities expressions for the required variables. The above-mentioned rules for basic operators should be followed while solving these inequalities.

If it is required to find the solution, first write the inequality as an equation and then solve the equation for the desired variable and obtain the required value.

The solution for the variable is less than or greater than the obtained value if there is a **strict **inequality. Whereas the solution is less than or equal to or greater than or equal to the value when there is **not a **strict inequality.

Finally represent the solution on the number line. Then draw the** open dot **at the endpoint for the **excluded **value of the solution and for the **included** value draw the **closed **dot.

### Linear Inequality With Two Variables

Linear inequalities in two variables show the inequality between two algebraic expressions that involve** distinct** variables. The solution to these inequalities is the values of ‘x’ and ‘y’ usually written in **ordered **pairs as (x,y).

These ordered pairs contain those values for which the given inequality stands** true** for both variables. The linear inequality in two variables is solved in the same manner as it is solved in one variable and according to the rules for basic arithmetic operators.

## Solved Examples

In order to understand the working of the tool, we need to solve some problems and analyze their result. So let’s review the problems solved by this exceptional tool.

### Example 1

Tyler wants to buy a suit of cost **$185**. He has a total savings of **$31** and he earns **$7** per hour from his job. Calculate the number of hours he has to work to collect the amount equal to the price of the suit.

This problem can be written in the form of expression as follows:

**7h + 31 $\ge$ 185**

Here the variable is hours and is represented as **‘h.’**

### Solution

The solution to the above problem by the calculator is given below.

#### Inequality Plot

Figure 1 shows the plot for the inequality in the x-y plane.

#### Result

After solving the inequality, some values from the obtained interval of the unknown variable are given below.

**h = 22, h = 23, h = 24, h = 25**

#### Interval Notation

The proper notation for the interval of the unknown variable ‘**h**’ is given below:

**[ 22, + ****$\infty$)**

#### Alternate Form

The solution can also be written in the form of inequality.

**h $\ge$ 22**

So Tyler has to work for at least **22** hours to buy the suit.

#### Number Line

#### The interval can be plotted in a single plane for better understanding which is shown in figure 2.

### Example 2

A maths student appears in an exam. He is asked to solve the following inequality and find the proper interval notation for the variable **‘x.’**

**– 3x – 7 < x + 9**

### Solution

According to the given expression, the calculator gives the following answer.

#### Inequality Plot

Both algebraic terms of inequality are separately drawn as a line in the cartesian plane in figure 3.

#### Result

The solution for variable **‘x’** is given as:

**x > – 4**

#### Interval Notation

The interval notation is provided below.

**(- 4, – ****$\infty$)**

#### Alternate Form

The alternate form for the resultant interval are given below:

**x > – 4**

**x + 4 > 0**

#### Number Line

Figure 4 illustrates the interval as a number line.