Absolute value
In this section, you will learn absolute value from the basics: from a simple definition, through the number line, to exam-style equations and inequalities.
What is absolute value?
The absolute value of a number tells how far that number is from zero on the number line. Direction does not matter; only distance matters.
That is why a positive number stays the same, while a negative number changes sign inside absolute value. The result is never negative.
Key notation
|5| = 5 and |-5| = 5
Section plan
What will you find on the subpages?
Work through the topic in a comfortable order, or jump straight to the part you need now.
Theory
Definition and properties
Learn the notation |x|, the piecewise definition, basic properties, and simple calculations.
Geometry
Geometric interpretation
See absolute value as distance on the number line: from zero or from a chosen number.
Equations
Absolute value equations
Learn how to solve equations such as |x - a| = r and tasks with several absolute values.
Inequalities
Absolute value inequalities
Practice turning inequalities into intervals and marking solutions on the number line.
Methods
Solving methods
Check when to use the definition, the geometric interpretation, or interval splitting.
What will you learn in this section?
Recognize absolute value as distance
You will understand why |x| cannot be negative and how to read expressions such as |x - 3|.
Split problems into cases
You will learn when the expression inside absolute value stays unchanged and when it changes sign.
Solve equations and inequalities
You will practice tasks from simple examples to situations with several intervals.
Choose the right method
You will see when geometric thinking is faster and when the definition is safer.
Most common uses
Distance on the number line
The expression |x - a| tells how many units separate x from the number a.
Equations with two answers
From |x| = 4 we get x = -4 or x = 4, because both points are equally far from zero.
Intervals in inequalities
The inequality |x - 2| < 5 describes numbers closer than 5 units from 2.
Where does absolute value appear?
Here are short examples that show common exam intuitions.
Evaluating a value|-8| = 8
Distance from a point|x - 3| = 2 means x = 1 or x = 5
Inequality as an interval|x| <= 4 means -4 <= x <= 4
Splitting into cases|x + 1| requires checking the sign of x + 1
Tip
Ask about distance first
If a task can be read as distance from a point, the number line is often faster than long calculations.
Worth noticing
Absolute value hides the sign
The numbers -7 and 7 have different signs, but the same absolute value. Absolute value remembers distance, not the side of the line.