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Absolute Value Calculator

Enter any real number or function and the calculator will instantly determine its absolute value.

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Absolute Value Equation Calculator

Our Absolute Value Calculator helps you find the absolute values of numbers, expressions, and functions. It also solves absolute value equations and inequalities, providing a clear step-by-step explanation for each solution.

What is the Absolute Value?

In mathematics, the absolute value (or modulus) of a real number x, written as |x|, is the non-negative value of x regardless of whether it is positive or negative. In other words, it measures the "distance" of a number from zero on the number line.

|x| = x, if x ≥ 0
|x| = -x, if x < 0 (here -x becomes positive)
|0| = 0

Graph of Absolute Value Functions

The general form of an absolute value function is:

F(x) = k + a |x – h|

Where:

h – Horizontal shift (moves the graph left or right)
k – Vertical shift (moves the graph up or down)
a – Stretch or shrink factor. If a > 0, the graph opens upwards; if a < 0, it opens downwards

Absolute Values Calculator

How to Solve Absolute Value Equations

Solving absolute value equations is similar to solving linear equations but requires handling the positive and negative cases separately. Here’s the step-by-step process:

Remember that |x| always represents a non-negative number.
The expression inside the absolute value can be any real number or function.
Check the right-hand side of the equation. If it is negative, there is no solution.
If the right-hand side is zero or positive, split the equation into two cases:
- Case 1: The expression inside the absolute value equals the positive value
- Case 2: The expression inside the absolute value equals the negative value
Solve both cases to find all possible solutions.

Example:

Solve the absolute value equation: -|x| = -9

Solution:

At first glance, it may seem that this equation has no solution because the right-hand side is negative. However, we need to handle the negative coefficient carefully:

The absolute value |x| is always non-negative.
Divide both sides by -1 to isolate |x|:

-|x| = -9
-1 × |x| = -9
(|x| × -1) / -1 = -9 / -1
|x| = 9

Now that we have |x| = 9, split into two cases:

|x| = a ⇒ x = +a or x = -a

Thus, |x| = 9 ⇒ x = 9 or x = -9

How the Absolute Value Calculator Works

Our absolute value equation calculator simplifies absolute value equations and inequalities in a few simple steps:

Input:

Enter a real number, variable expression, or equation in the input field.
Click "Calculate" to get the solution.

Output:

The calculator provides the solution(s) to the absolute value equation or inequality.
If needed, it splits the equation into positive and negative cases and solves each separately.
For inequalities, it shows the solution intervals on the number line.

Applications of Absolute Values

Absolute values are used in various mathematical and real-life contexts, including:

Measuring distance on a number line
Solving equations and inequalities
Defining piecewise functions
Error analysis and deviations in statistics

Reference:

From Wikipedia: Absolute value function – Definition and properties

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