Digital SAT Math · Algebra

Absolute value equations

Absolute value is distance on the number line, not a fancy pair of bars. The equation $|x - c| = d$ says nothing more than “$x$ is $d$ units away from $c$” — so the answers sit at $c + d$ and $c - d$, when those make sense. Almost every miss on this skill is either splitting before the bars are alone, accepting a negative distance, or drawing a “greater than” solution as a closed band between the endpoints instead of two rays going outward.

On the test

Recognition cues: vertical bars around an expression; the words distance, at least / at most paired with a deviation from a center; a number line with a segment or two opposite rays; and any stem that asks whether a solution set is empty.

Pattern recognition

Five shapes cover nearly everything:

1. Distance equation — $|x - c| = d$ or $|ax + b| = k$ with $k \ge 0$. Two roots (or one if $k = 0$).
2. Isolate, then split — bars sit inside a larger equation; clear the outside first so the right-hand side is a single number.
3. No solution — after isolating, the absolute value equals a negative number, or a “less than negative” inequality asks for the empty set.
4. Inequality band or rays — “less than” is between the endpoints; “greater than” is outside them.
5. Check both cases — the other side of the equation still has the variable; each case must be substituted back.

Method

1. Isolate the absolute value. Add, subtract, multiply, or divide until the bars are alone on one side. Do not open the bars until that is done.
2. Read the isolated value. If it is negative, stop: no real solution (an absolute value cannot equal a negative number). If it is zero, there is exactly one solution. If it is positive, continue.
3. Split into two linear cases (equations) or rewrite as a compound inequality (inequalities):
   • $|A| = k$ with $k > 0$ → $A = k$ or $A = -k$
   • $|A| < k$ (or $\le$) with $k > 0$ → $-k < A < k$ (between band)
   • $|A| > k$ (or $\ge$) with $k > 0$ → $A < -k$ or $A > k$ (two outward rays)
4. Solve each linear piece with ordinary one-variable algebra.
5. Check when the other side has a variable, or whenever a case looked suspicious. Reject any root that fails the original equation.
6. Match the ask. One root, both roots, their sum or product, a count of solutions, a compound inequality, or a graph — finish the question that was asked.

Practice

Answer first, then open the explanation. Two items are student-produced response (type a single number). Three items depend on a number line — read whether the endpoints are filled or hollow, and whether the shading runs between them or outward, before you touch the choices. Every wrong choice below is a specific error, not filler.

Common mistakes

1. Splitting before isolating — opening $|2x - 1| - 5 = 2$ into $2x - 1 - 5 = 2$ and its negative twin. Clear the outside first so the bars stand alone.
2. Accepting a negative absolute value — treating $|A| = -3$ as two ordinary linear equations and reporting two roots that fail the original.
3. Drawing the greater-than set as a band — sketching $|x - c| \ge d$ as the segment from $c - d$ to $c + d$. That segment is the less-than solution.
4. Losing endpoint style — open vs closed circles (or $<$ vs $\le$) flipped when translating a graph or a compound inequality.
5. Forgetting the center — solving $|x - 3| = 5$ as $x = \pm 5$ instead of $3 \pm 5$.
6. Keeping an extraneous root — when the other side of the equation has $x$, a case can produce a number that makes that side negative or simply fails substitution.
7. Partial sign on the negative case — writing $-(2x + 1)$ as $-2x + 1$ instead of $-2x - 1$.
8. Reporting one root when the ask wanted both — or reporting $x$ when the stem asked for the sum or product of the solutions.
9. Treating $|A| < 0$ as a band — a strict less-than against a negative (or zero) number is empty (or at most a single point for $\le 0$).

FAQ

Can an absolute value equal zero? Yes. $|A| = 0$ means $A = 0$, exactly one solution. Zero is allowed; negative is not.

Do I always get two solutions? No. You get two when the isolated right-hand side is positive, one when it is zero, and none when it is negative. Inequalities can also yield a whole band, two rays, or the empty set.

Is $|x| > -3$ all real numbers? Yes — every absolute value is at least 0, which is already greater than $-3$. The trap twin is $|x| < -3$, which is empty. Read the inequality direction before you celebrate.

Should I check every root? Check when the other side of the equation still contains the variable, when you isolated through a messy chain of signs, or when a root looks like it might make an absolute-value-equals-linear identity fail. Pure $|ax+b| = k$ with $k > 0$ and no extra variable rarely needs a check beyond a 5-second substitute.

Editorial team · last reviewed 2026-08-12