# Inequality Calculator with steps | Inequality solver

## Inequality Calculator

Enter a mathematical expression... #### Oh snap!

An error occurred while processing this operation.

Welcome to our Inequality Calculator! This powerful tool allows you to easily solve any inequality with just a few simple steps. Simply enter the inequality into the provided input field and hit the “Calculate” button. The Inequality Calculator will then provide you with a step-by-step solution.

Whether you’re a student trying to ace your math exams or a professional looking for a quick and accurate way to solve inequalities, our Inequality Calculator is the perfect tool for you. Give it a try now and see just how useful it can be!

Valid functions and symbols Description
sqrt() Square root
ln() Natural logarithm
log() Logarithm base 10
^ Exponents
abs() Absolute value
sin(), cos(), tan(), csc(), sec(), cot() Basic trigonometric functions
asin(), acos(), atan(), acsc(), asec(), acot() Inverse trigonometric functions
sinh(), cosh(), tanh(), csch(), sech(), coth() Hyperbolic functions
asinh(), acosh(), atanh(), acsch(), asech(), acoth() Inverse hyperbolic functions
pi PI number (π = 3.14159...)
e Neper number (e= 2.71828...)
i To indicate the imaginary component of a complex number.

Table 1: Valid functions and symbols

## Inequality definition | What is an inequality in math?

In math, an inequality is a statement that compares the value of one expression to the value of another using one of the following inequality symbols:

• Less than: <
• Less than or equal to: ≤
• Greater than: >
• Greater than or equal to: ≥

Inequalities are used to describe situations where one value is not equal to another value. They are often used in algebra to describe conditions that must be met for a solution to be valid.

For example:

• x + 3 < 5 is an inequality that says “x + 3 is less than 5”
• y ≥ 10 is an inequality that says “y is greater than or equal to 10”

The solution to an inequality is a set of values that make the inequality true. For example, the solution to the inequality x + 3 < 5 is the set of all values of x that are less than 2 (since 5 – 3 = 2). The solution to the inequality y ≥ 10 is the set of all values of y that are greater than or equal to 10.

## Types of Inequalities

There are several types of inequalities:

1. Linear inequalities: These are inequalities that involve only one variable and can be represented in the form “ax + b < c” or “ax + b > c”, where a, b, and c are constants and x is the variable. An example of a linear inequality is “2x + 3 < 7”.

2. Quadratic inequalities: These are inequalities that involve a variable raised to the second power, such as “x2 + 2x + 1 < 0″. Quadratic inequalities can be solved by finding the values of x that make the inequality true and then testing those values to determine which ones are valid solutions.

3. Absolute value inequalities: These are inequalities that involve the absolute value of a variable, such as “|x – 3| < 4”. Absolute value inequalities can be solved by splitting them into two separate inequalities and solving each one separately.

4. Rational inequalities: These are inequalities that involve rational expressions, such as “1/x < 2”. Rational inequalities can be solved by finding the values of x that make the inequality true and then testing those values to determine which ones are valid solutions.

These are just a few examples of the types of inequalities that exist. There are many other types of inequalities that can be used in different mathematical contexts and in solving problems.

## How to solve inequalities

### Solving linear inequalities

Most techniques for solving linear equations are applicable to the calculation of linear inequalities. Therefore, in order to find the solution to a real inequality, you can add or subtract any real number to both sides of an inequality, and you can also multiply or divide both sides by any positive real number to create equivalent inequalities.

To illustrate the above explained below I present how we can solve the following linear inequality:

5x+3x8>3

Step 1: Simplify both sides of the inequality.
8x8>3
Step 2: Add 8 to both sides.
8x8+8>3+8
8x>11
Step 3: Divide both sides by 8.
 8x 8
>
 11 8
x>
 11 8

Solution:
x>
 11 8

1. Write the quadratic inequality in standard form, eg: Ax2+Bx+C>0
2. Determine the critical points: to do so, find the solutions of the related quadratic equation.
3. Use the critical points to determine the intervals where the inequality is correct. Write the solution in interval notation.

Example: Solve the quadratic inequality x2+5x-2>0 ### How to solve absolute value inequalities

1. To solve an absolute value inequality, you will need to split the inequality into two separate inequalities and solve them separately.
• If the inequality has a greater than symbol, >, create two inequalities as follows:

(expression inside absolute value) <- (number on the right side)

(expression inside absolute value) > (number on the right hand side).

The same setting is used for a greater than or equal to sign, >=.

• If the symbol of the inequality is “less than”, <, set up two inequalities as follows:

(expression within absolute value) < (number on the right hand side)

(expression within absolute value) > – (number on the right hand side).

The same setting is used for inequality having a less than or equal to sign, <=.

1. Then solve the inequalities created. The solution to the absolute value inequality is the union of the solutions.

Let’s see an example of how to solve an absolute value inequality :

Let's solve the inequation
|5x8|3

Solve Absolute Value.
|5x8|3
We know either5x83or5x83

5x83(A)
5x11
 5x 5
 11 5
(Divide both sides by 5)
x
 11 5

5x83(B)