Opposites are two numbers that are the same distance from zero on the number line, with one negative and one positive. For instance, h is the opposite of - h.
We can represent - h (read: “the opposite of h,” or “negative h”) with a Circle of Evaluation:
Absolute value is the (positive) distance of a number from zero. We annotate absolute value like this: |h|, with h being any given number.
When we encounter an expression like |h|, we say "the absolute value of h."
We can represent |h| with a Circle of Evaluation:
Because opposites are the same distance away from zero, they will always have the same absolute value. So, |4| = 4 and | - 4| = 4.
The algebraic expressions |h| and - h sometimes produce the same outcome, and they sometimes produce different outcomes. |h| is always positive or zero, while - h can be negative, zero, or positive.
We can also create expressions that utilize both opposite and absolute value. For instance:
-
We can find the opposite of an absolute value: - | x |
-
We can find the absolute value of an opposite. | - x |
Thinking about the structure of the expression (and studying its Circle of Evaluation) can help us understand if it is positive or negative.
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.