📖Definition
Absolute value is the distance a number is from zero on the number line, regardless of direction. It is always zero or positive, and is written with two vertical bars, as in |x|.
📐Formula
If a number is already positive or zero, its absolute value is itself. If it is negative, its absolute value is the positive version of that number.
📝Step-by-Step Guide
Read absolute value as distance
Think of |x| as "how far is x from zero", not as "remove the minus sign".
Evaluate a numerical expression
Simplify what is inside the bars first, then apply the absolute value.
Solve an absolute value equation
An equation like |x − a| = b splits into two cases: x − a = b and x − a = −b.
Solve both cases and check
Solve each resulting equation separately, then substitute both answers back in to confirm they work.
🧭Absolute value as distance
On a number line, both 7 and −7 are the same distance from zero, so |7| and |−7| are both equal to 7.
This distance interpretation is why absolute value equations usually have two solutions: two different starting points can both be the same distance from a target.
🧭Absolute value in real life
Absolute value shows up whenever only the size of a difference matters, not its direction. Temperature differences, error margins, and elevation changes above or below sea level are common examples.
⚠️Common Mistakes to Avoid
- Treating absolute value as simply deleting the minus sign without checking the result is non-negative
- Only finding one solution to an absolute value equation instead of two
- Forgetting to simplify inside the bars before applying the absolute value
- Assuming |x| can be negative
- Mixing up |x − a| = b with |x| − a = b, which are different expressions
✏️Practice Problems
Evaluate |−7|
💡 Hint: Absolute value is always non-negative.
Answer: 7
Evaluate |5|
💡 Hint: A positive number stays the same.
Answer: 5
Solve |x| = 7
💡 Hint: There are two numbers whose distance from zero is 7.
Answer: x = 7 or x = −7
Evaluate |3 − 10|
💡 Hint: Simplify inside the bars first.
Answer: 7
Solve |x − 3| = 5
💡 Hint: Split into two cases: x − 3 = 5 and x − 3 = −5.
Answer: x = 8 or x = −2
Solve |x| + 2 = 9
💡 Hint: Isolate the absolute value first.
Answer: x = 7 or x = −7
Printable Absolute Value practice
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❓Absolute Value FAQ
Can absolute value ever be negative?
No. Absolute value measures a distance, and distances are never negative. The smallest possible value is 0, when the number itself is 0.
Why do absolute value equations have two solutions?
Because two different numbers, one positive and one negative, can be the same distance from zero (or from any other reference point).
What is the modulus symbol?
The two vertical bars, |x|, are read as "the absolute value of x" in most countries and "the modulus of x" in others. Both mean exactly the same thing.