Mathématiques

What Absolute Value Means

The bars mean distance from zero, so |−7| = 7 and |3 − 9| = 6. On the Digital SAT that definition is why |x − 4| = 6 splits into two equations.

Absolute value

The absolute value of a number is its distance from zero on the number line, so it is never negative: |−7| and |7| are both 7.

Aussi appelé: modulus
Où les élèves le rencontrent: US middle-school math and then routinely on the Digital SAT; GCSE mathematics in England does not use the notation, so students moving between systems often meet the bars for the first time in SAT preparation.

La réponse en bref
The absolute value of a number is how far it is from zero, written between two vertical bars: |−7| = 7 and |7| = 7. Because it measures a distance, it is never negative. On the Digital SAT, an equation such as |x − 4| = 6 splits into x − 4 = 6 and x − 4 = −6.

Un exemple

|3 − 9| = |−6| = 6, but |3| − |9| = 3 − 9 = −6

Two expressions that differ by the position of the bars and give opposite answers. On the left the subtraction happens inside, so the bars measure how far −6 is from zero. On the right each number is measured separately and the results are subtracted afterwards. The bars group what they enclose, exactly as brackets do.

Distance, not a rule about deleting minus signs

Make it positive is the shortcut most students are handed, and it works right up to the point where something other than a plain number sits between the bars. Then it stops working, because |x| is not x with the sign removed — it is the distance from x to zero, and until you know whether x is positive there is nothing to remove.

Reading it as distance also unlocks the version that actually appears in test questions. |a − b| is the distance between a and b on the number line, so a sentence like the temperature stayed within 3 degrees of 20 becomes |t − 20| ≤ 3 without any further thought. That translation is worth more marks than the sign rule ever was.

Everything inside the bars happens first

The bars are grouping symbols. Whatever is between them is worked out completely, and only then is the distance taken. |3 − 9| is |−6|, which is 6.

The mistake to watch for is splitting the expression across the bars, which turns one calculation into a different one: |3| − |9| is 3 − 9, which is −6. Same digits, opposite answer. If an expression inside the bars contains a subtraction, finish it before touching the bars at all.

Why these equations have two answers

|x − 4| = 6 is asking which numbers sit 6 away from 4. Going up gives 10, going down gives −2, and both are genuine solutions. That is not a rule to memorise, it is the definition read aloud, and it is why the standard move is to write x − 4 = 6 and x − 4 = −6 and solve each.

The same reading tells you when there is no solution at all. |x| = −3 asks for a number whose distance from zero is negative, and distance cannot be negative, so nothing satisfies it. A student who has learned the two-equation move as a ritual will happily produce two answers here; one who has learned the definition will not.

Questions fréquentes

Is absolute value the same as modulus?

Yes. Modulus is the older British term and absolute value the American one, and both refer to the same two vertical bars and the same meaning. UK textbooks that cover the topic tend to say modulus; the College Board and US curricula say absolute value. Nothing changes but the word.

Can an absolute value be negative?

No, never. It is a distance, and distances start at zero. What can be negative is an absolute value with something in front of it: −|4| is −4, because the minus sign is applied after the distance has been taken. |x| itself is always 0 or more, whatever x turns out to be.

Does the SAT test absolute value?

It appears, but not as a topic of its own. The SAT Math section is built from four content domains — Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry — and absolute value turns up inside them, most often in linear equations and inequalities, with no explanation offered.

Why does |x − 4| = 6 have two solutions?

Because two different numbers are 6 units from 4: one above it at 10 and one below it at −2. The equation asks a question about distance, and distance does not record direction, so both journeys satisfy it. Checking both in the original equation confirms they work.

Sources

The Math Section — College Board

Dernière mise à jour 21 août 2026