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What Is the Difference Between Absolute Value & Opposites?

A number line displays numbers in their correct sequential order. Number lines are helpful when learning the relationships between numbers. The line is divided around 0 at the center. To the left of 0 are negative numbers, and to the right of 0 are positive numbers. All of the numbers on the line, positive and negative and 0, form the definition of integers. The definition of integers and of the number line play into the definitions of absolute value and opposite numbers.

Absolute Value Definition
- On a number line, -5 and 5 are both 5 notches away from 0, just in different directions. In some situations, the distance traveled is more important than the direction (or sign). For problems where sign is of no importance, absolute value is used. Denoted by a set of vertical bars, absolute value states that whatever is between the bars should be treated as a positive number. For example, |-6| = 6 since the only concern is that it is 6 spaces from 0.
Opposite Numbers
- On the number line, 0 separates the two sides of opposite numbers. The positive integers on the right each have an opposing negative integer on the left. Adding an integer to its opposite always results in 0. For example, 3 + -3 = 0.
Opposite Absolute Values
- Can absolute value integers have an opposite? No, but they can have a negative version. For example, - |-5| = -(5) = -5. Note that while the answer of an absolute value with a negative sign in front may sometimes look like the opposite of whatever was inside, it would turn out the same answer no matter what the internal sign was. Thus - |-4| = -4 and -|4| = -4.
Solving Equations
- Equations may contain a variable within the absolute value sign. While most operations can be eliminated by applying its opposite operation, absolute value lacks an opposite. Therefore, it must be solved for both the positive and the negative versions of the interior expression because either is equally likely.
  
  The equation |x + 3| = 6 is solved by creating the equations (x + 3) = 6 and -(x + 3) = 6 and solving for both values of "x". In the first equation, simply subtract 3 from both sides: x = 3. Distribute the negative in the second equation: -x - 3 = 6. Add 3 to both sides: -x = 9. Divide both sides by -1: x = -9. Thus, the answers are x = 3 and x = -9.

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