Absolute value
Also known as: modulus
Absolute value is a number's distance from zero on the number line, written with vertical bars: |x|. Because distance is never negative, absolute value is always zero or positive — |−5| and |5| both equal 5.
Absolute value measures how far a number is from zero on the number line, regardless of direction. It is written with vertical bars: |x|. Since distance cannot be negative, absolute value is always zero or positive — |7| = 7, |−7| = 7, and |0| = 0. For a negative number, taking the absolute value simply drops the sign.
Absolute value equations usually have two solutions, because two different numbers sit the same distance from zero. If |x| = 4, then x = 4 or x = −4. The same logic extends to expressions: solving |x − 3| = 5 means x − 3 = 5 or x − 3 = −5, giving x = 8 or x = −2. Geometrically, |x − 3| = 5 asks which numbers are exactly 5 units from 3.
Inequalities follow two patterns worth memorizing. A "less than" inequality traps values in a band: |x| < 4 means −4 < x < 4. A "greater than" inequality splits into two rays: |x| > 4 means x > 4 or x < −4. Graphs of absolute value functions like y = |x| form a V shape, which shifts around the plane just like a parabola in vertex form.
The ACT, SAT, and GRE all test absolute value. Common questions include solving two-case equations and inequalities, evaluating nested expressions like |3 − 7| − |2 − 9|, and interpreting absolute value as distance on the number line.
Key takeaways
- Absolute value is distance from zero on the number line, so it is never negative.
- |x| = a (for positive a) has two solutions: x = a and x = −a.
- |x| < a traps x between −a and a; |x| > a splits into x > a or x < −a.
- The graph of y = |x| is a V shape, and |x − h| represents distance from h.
- The ACT, SAT, and GRE test absolute value equations, inequalities, and distance interpretations.