Absolute value tells how far a number is from 0 on the number line. Distance is never negative, so an absolute value is always 0 or positive. This idea helps students compare numbers, measure differences, and understand symmetry around 0.
For example, -5 and 5 have the same absolute value because both are 5 units from 0.
The symbol for absolute value is two vertical bars, such as |x|. To evaluate it, find the distance of the number or expression inside the bars from 0. On a number line, numbers that are opposites appear the same distance from 0 but on different sides.
Absolute value is also used in equations, inequalities, error measurement, and real-world situations where only the size of a difference matters.
Key Facts
- |x| means the distance of x from 0 on the number line.
- |x| ≥ 0 for every real number x.
- |5| = 5 and |-5| = 5 because both numbers are 5 units from 0.
- |0| = 0 because 0 is zero units from itself.
- If x ≥ 0, then |x| = x.
- If x < 0, then |x| = -x, which makes the result positive.
Vocabulary
- Absolute value
- The distance of a number from 0 on the number line.
- Number line
- A straight line used to show numbers in order and compare their positions.
- Opposites
- Two numbers that are the same distance from 0 but on opposite sides of the number line.
- Nonnegative
- A number that is greater than or equal to 0.
- Expression
- A mathematical phrase made of numbers, variables, and operations.
Common Mistakes to Avoid
- Making an absolute value negative, such as writing |-8| = -8, is wrong because absolute value represents distance and distance cannot be negative.
- Changing every sign inside the bars automatically is wrong because you must first evaluate the expression inside, such as |-3 + 10| = |7| = 7.
- Thinking |a| always equals a is wrong because this is only true when a is 0 or positive, while |-a| depends on the value of a.
- Ignoring the order of operations with absolute value is wrong because absolute value acts like grouping symbols, so simplify inside the bars before applying the absolute value.
Practice Questions
- 1 Evaluate: |-12|, |9|, and |0|.
- 2 Evaluate: |7 - 15| + |-4|.
- 3 Explain why -6 and 6 have the same absolute value even though they are different numbers.