Learn on PengiPengi Math (Grade 6)Chapter 1: Rational Numbers — Whole Numbers, Fractions, and Decimals

Lesson 5: Absolute Value as Distance from Zero

In this Grade 6 Pengi Math lesson from Chapter 1 on Rational Numbers, students learn to define absolute value as a number's distance from zero on the number line and apply that concept to real-world contexts. The lesson also covers comparing absolute values, distinguishing magnitude from order, and using absolute value to calculate the distance between two points.

Section 1

Defining Absolute Value as Distance from Zero

Property

The absolute value of a number is its distance from zero on the number line.
The absolute value of a number $n$ is written as $|n|$ and $|n| \geq 0$ for all numbers.
Absolute values are always greater than or equal to zero.

Examples

The absolute value of $-9$ is $9$ , because $-9$ is $9$ units away from $0$ . This is written as $|-9| = 9$ .
The absolute value of $25$ is $25$ , because $25$ is $25$ units away from $0$ . This is written as $|25| = 25$ .
The equation $|x| = -4$ has no solution. Absolute value represents distance, which cannot be a negative number.

Explanation

Think of absolute value as a 'distance-meter' from zero. Since distance can't be negative, the absolute value of any number, positive or negative, will always be a positive result or zero. It simply tells you how far away you are.

Section 2

Absolute value

Property

The absolute value of $x$ is defined by

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

Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.

Examples

To evaluate $|-9|$ , since $-9 < 0$ , we use the second case of the definition: $|-9| = -(-9) = 9$ .

To evaluate the expression $10 - |5 - 8|$ , first compute the operation inside the absolute value bars: $5 - 8 = -3$ . Then take the absolute value: $|-3| = 3$ . Finally, subtract: $10 - 3 = 7$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Absolute Value as Distance from Zero

Property

Examples

The absolute value of $-9$ is $9$ , because $-9$ is $9$ units away from $0$ . This is written as $|-9| = 9$ .
The absolute value of $25$ is $25$ , because $25$ is $25$ units away from $0$ . This is written as $|25| = 25$ .
The equation $|x| = -4$ has no solution. Absolute value represents distance, which cannot be a negative number.

Explanation

Section 2

Absolute value

Property

The absolute value of $x$ is defined by

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

Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.

Examples

To evaluate $|-9|$ , since $-9 < 0$ , we use the second case of the definition: $|-9| = -(-9) = 9$ .

To evaluate the expression $10 - |5 - 8|$ , first compute the operation inside the absolute value bars: $5 - 8 = -3$ . Then take the absolute value: $|-3| = 3$ . Finally, subtract: $10 - 3 = 7$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Defining Absolute Value as Distance from Zero

Property

Examples

The absolute value of $-9$ is $9$ , because $-9$ is $9$ units away from $0$ . This is written as $|-9| = 9$ .
The absolute value of $25$ is $25$ , because $25$ is $25$ units away from $0$ . This is written as $|25| = 25$ .
The equation $|x| = -4$ has no solution. Absolute value represents distance, which cannot be a negative number.

Explanation

Section 2

Absolute value

Property

The absolute value of $x$ is defined by

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

Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.

Examples

To evaluate $|-9|$ , since $-9 < 0$ , we use the second case of the definition: $|-9| = -(-9) = 9$ .

To evaluate the expression $10 - |5 - 8|$ , first compute the operation inside the absolute value bars: $5 - 8 = -3$ . Then take the absolute value: $|-3| = 3$ . Finally, subtract: $10 - 3 = 7$ .