Absolute Value

Arithmetic
operation

Also known as: magnitude, distance from zero, absolute-value-inequalities, absolute-value-function

Grade 6-8

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The distance of a number from zero on the number line, always non-negative; written |x|. Used for distances, errors, and tolerances where direction doesn't matter.

Definition

The distance of a number from zero on the number line, always non-negative; written |x|. For any real number, absolute value strips away the sign and returns only the magnitude.

💡 Intuition

-5 and 5 are both 5 units from zero, so |-5| = |5| = 5.

🎯 Core Idea

Absolute value strips away the sign, leaving only magnitude.

Example

|-7| = 7 (distance 7 from zero), |3| = 3, |0| = 0; also |5 - 8| = 3.

Formula

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

Notation

|x| means the absolute value of x

🌟 Why It Matters

Used for distances, errors, and tolerances where direction doesn't matter.

💭 Hint When Stuck

Draw a number line and count the steps from the number to zero -- that count is the absolute value.

Formal View

|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}, \quad \text{equivalently } |x| = \sqrt{x^2}

🚧 Common Stuck Point

Confusing |-x| with -|x|: |-3| = 3 but -|{-3}| = -3. Always non-negative inside.

⚠️ Common Mistakes

  • Thinking |-a| always equals -a — it equals the positive magnitude regardless of input sign
  • Incorrectly distributing absolute value across operations — |a + b| \neq |a| + |b| in general
  • Forgetting that absolute value equations like |x| = 5 have two solutions: x = 5 and x = -5

Frequently Asked Questions

What is Absolute Value in Math?

The distance of a number from zero on the number line, always non-negative; written |x|. For any real number, absolute value strips away the sign and returns only the magnitude.

What is the Absolute Value formula?

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

When do you use Absolute Value?

Draw a number line and count the steps from the number to zero -- that count is the absolute value.

Prerequisites

integers

How Absolute Value Connects to Other Ideas

To understand absolute value, you should first be comfortable with integers. Once you have a solid grasp of absolute value, you can move on to distance formula and inequalities.

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