Ekuation

Absolute Value Calculator

A simple and precise absolute value calculator that helps you find the absolute value (magnitude) of any number or expression. The absolute value of a number represents its distance from zero on the number line, regardless of whether it is positive or negative.

Enter any positive or negative number

What is Absolute Value?

The absolute value of a number is its distance from zero on a number line, regardless of whether the number is positive or negative. It's denoted by vertical bars on both sides of a number or expression, like this: ∣x∣|x|.

The absolute value function returns the positive version of any number. You can think of it as removing the negative sign from negative numbers, while leaving positive numbers unchanged.

Formal Definition

The absolute value of a real number xx is defined as:

∣x∣={x,if x≥0−x,if x<0|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

For example, ∣5∣=5|5| = 5 and ∣−5∣=−(−5)=5|-5| = -(-5) = 5.

Key Properties of Absolute Value

  • ∣x∣≥0|x| \geq 0 for any real number xx
  • ∣x∣=0|x| = 0 if and only if x=0x = 0
  • ∣−x∣=∣x∣|-x| = |x| for any real number xx
  • ∣x⋅y∣=∣x∣⋅∣y∣|x \cdot y| = |x| \cdot |y| for any real numbers xx and yy
  • ∣xy∣=∣x∣∣y∣\left|\frac{x}{y}\right| = \frac{|x|}{|y|} for any real numbers xx and y≠0y \neq 0
  • ∣x+y∣≤∣x∣+∣y∣|x + y| \leq |x| + |y| (Triangle Inequality)

Examples

Basic Examples

  • ∣5∣=5|5| = 5
  • ∣−7∣=7|-7| = 7
  • ∣0∣=0|0| = 0
  • ∣3.14∣=3.14|3.14| = 3.14
  • ∣−2.5∣=2.5|-2.5| = 2.5

Expressions

  • ∣x−3∣|x - 3| represents the distance between xx and 33 on a number line
  • ∣2x+5∣|2x + 5| is the absolute value of the expression 2x+52x + 5
  • ∣x2−4∣|x^2 - 4| is always non-negative regardless of the value of xx

Real-World Applications

Distance Calculation

Absolute value is commonly used to calculate distance. For example, the distance between two points aa and bb on a number line is ∣a−b∣|a - b|.

Error Measurements

In statistics and data analysis, absolute value is used to calculate absolute error: ∣actual−measured∣|actual - measured|.

Financial Calculations

Absolute value helps analyze financial gains or losses without considering whether they are positive or negative, focusing only on magnitude.

Scientific Measurements

When scientists need to measure deviation from a standard or expected value, absolute value provides the magnitude of the deviation regardless of direction.

Solving Absolute Value Equations

An absolute value equation is an equation that contains an absolute value expression. To solve such equations, we need to consider both possible cases:

For an equation of the form ∣expression∣=a|expression| = a (where a>0a > 0):

We need to solve two separate equations:

expression=a or expression=−aexpression = a \text{ or } expression = -a

Example:

To solve ∣x−3∣=4|x - 3| = 4:

  • Case 1: x−3=4x - 3 = 4, so x=7x = 7
  • Case 2: x−3=−4x - 3 = -4, so x=−1x = -1

Therefore, the solutions are x=7x = 7 and x=−1x = -1.

Absolute Value Inequalities

Absolute value can also be used in inequalities:

For ∣expression∣<a|expression| < a (where a>0a > 0):

−a<expression<a-a < expression < a

Example:

∣x−3∣<2|x - 3| < 2 means −2<x−3<2-2 < x - 3 < 2

So 1<x<51 < x < 5

For ∣expression∣>a|expression| > a (where a>0a > 0):

expression<−a or expression>aexpression < -a \text{ or } expression > a

Example:

∣x−3∣>2|x - 3| > 2 means x−3<−2x - 3 < -2 or x−3>2x - 3 > 2

So x<1x < 1 or x>5x > 5

Tips for Working with Absolute Value

  • Remember that absolute value is always non-negative
  • When in doubt, consider the cases separately (positive and negative)
  • The expression ∣x−a∣|x - a| represents the distance between xx and aa on a number line
  • Be careful with absolute value inequalities—they often result in compound inequalities or disjoint solution sets

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