Lesson 5-2: Understand Additive Inverses

Property

Two numbers are opposites when they are the same distance away from zero, but in opposite directions.
For example, “3” represents the point that is 3 units to the right of 0, and “$-3$” is its opposite, three units to the left of 0.
Zero is its own opposite.

Examples

• The opposite of 9 is $-9$, as both numbers are 9 units away from 0.
• The opposite of the opposite of $-4$ is $-(-(-4)) = -4$. The first opposite is 4, and the opposite of 4 is $-4$.
• If earning 5 points in a game is represented by $+5$, its opposite is losing 5 points, represented by $-5$.

Explanation

Opposites are mirror images of each other across zero. They have the same distance from zero, just in different directions. Taking the opposite of an opposite brings you right back to where you started!

Property

Using a Number Line:

• To find the opposite of $x$: Graph $x$, count the number of units to $0$, and find the point that is the same distance on the opposite side of $0$.
• To find the absolute value $|x|$: Graph $x$ and count the total distance in units from $0$.

Examples

Property

The absolute value $|a|$ of an integer $a$ is the distance from the point on the number line to $0$. A number and its opposite have the same absolute value. Every integer has an opposite, or additive inverse. $0$ is its own opposite.

Examples