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Absolute Value Equations on the SAT

SAT Math · Algebra · Updated October 2026

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Absolute value is distance from $0$, so it is never negative. An equation like $|x - 3| = 5$ splits into two cases, $x - 3 = 5$ or $x - 3 = -5$, which gives two solutions, $8$ and $-2$.

Before you split, get the absolute value by itself on one side. If it then equals a negative number, there is no solution, and you can move on.

Once students know the two-case rule, these questions go quickly. The SAT also likes to describe $|x - a| \le b$ in words, as "within $b$ of $a$", in questions about tolerances and temperatures.

What to do when you see

Examples

Isolate, then split

Solve $3|x - 2| + 1 = 13$.

  1. Subtract $1$ and divide by $3$: $|x - 2| = 4$.
  2. Split: $x - 2 = 4$ or $x - 2 = -4$.
  3. $x = 6$ or $x = -2$.

Writing a tolerance

A thermostat keeps a room within $4$ degrees of $68$°F. Write this as an inequality for the temperature $t$.

  1. "Within $4$ of $68$" means the distance from $t$ to $68$ is at most $4$.
  2. $|t - 68| \le 4$.
  3. That is the same as $64 \le t \le 72$.

A variable on both sides

Solve $|2x - 6| = x + 3$.

  1. Case 1: $2x - 6 = x + 3$, so $x = 9$.
  2. Case 2: $2x - 6 = -(x + 3)$, so $3x = 3$ and $x = 1$.
  3. Check both: $|12| = 12$ and $|-4| = 4$. Both work.

Common mistakes

Splitting before isolating. In $3|x - 2| + 1 = 13$, get $|x - 2|$ alone first.

Only solving the positive case. Almost every absolute value equation has two solutions.

Solving an impossible equation. $|x + 4| = -2$ has no solution. Write "no solution" and move on.

Negating only part of the right side. The second case of $|2x - 6| = x + 3$ is $2x - 6 = -x - 3$.

Mixing up "within" and "more than". Within $3$ of $10$ is $|x - 10| \le 3$. More than $3$ away is $|x - 10| > 3$.

Your turn

Isolate the absolute value first in every question.

Easy
What are all the solutions to $|x| = 7$?
  • A$7$ only
  • B$-7$ only
  • C$-7$ and $7$
  • D$0$ and $7$
Show solution
Answer: C. Both $7$ and $-7$ are a distance of $7$ from $0$.
EasyStudent-produced response
What is the sum of all solutions to $|x - 3| = 5$?
Enter your answer.
Show solution
Answer: $6$. $x - 3 = 5$ gives $x = 8$, and $x - 3 = -5$ gives $x = -2$. The sum is $6$.
Medium
What are all the solutions to $|2x + 1| = 9$?
  • A$4$ only
  • B$-5$ and $4$
  • C$-4$ and $5$
  • D$-4$ and $4$
Show solution
Answer: B. $2x + 1 = 9$ gives $x = 4$. $2x + 1 = -9$ gives $2x = -10$ and $x = -5$. Choice A finds only the positive case.
Medium
How many solutions does the equation $|x + 4| = -2$ have?
  • AZero
  • BExactly one
  • CExactly two
  • DInfinitely many
Show solution
Answer: A. An absolute value is never negative, so it can never equal $-2$. There are no solutions.
MediumStudent-produced response
What is the greater solution to $3|x - 2| + 1 = 13$?
Enter your answer.
Show solution
Answer: $6$. $3|x - 2| = 12$, so $|x - 2| = 4$. Then $x = 6$ or $x = -2$. The greater solution is $6$.
Medium
Which inequality describes all numbers $x$ that are within $3$ units of $10$ on the number line?
  • A$|x + 10| \le 3$
  • B$|x - 3| \le 10$
  • C$|x - 10| \ge 3$
  • D$|x - 10| \le 3$
Show solution
Answer: D. The distance from $x$ to $10$ is $|x - 10|$, and within $3$ means at most $3$. Choice C describes numbers at least $3$ away.
HardStudent-produced response
What value of $x$ satisfies $|x - 5| = |x + 1|$?
Enter your answer.
Show solution
Answer: $2$. $x$ must be the same distance from $5$ as from $-1$, so it is halfway between them: $\dfrac{5 + (-1)}{2} = 2$. Check: $|{-3}| = 3$ and $|3| = 3$.
Hard
A thermostat keeps the temperature $t$ of a room, in degrees Fahrenheit, within $4$ degrees of $68$. Which inequality describes $t$?
  • A$|t + 68| \le 4$
  • B$|t - 68| \le 4$
  • C$|t - 4| \le 68$
  • D$|t - 68| \ge 4$
Show solution
Answer: B. The distance from $t$ to $68$ is at most $4$: $|t - 68| \le 4$, which means $64 \le t \le 72$.
HardStudent-produced response
What is the sum of all solutions to $|2x - 6| = x + 3$?
Enter your answer.
Show solution
Answer: $10$. Case 1: $2x - 6 = x + 3$ gives $x = 9$. Case 2: $2x - 6 = -x - 3$ gives $x = 1$. Both check in the original equation, so the sum is $10$.
HardStudent-produced response
The equation $|x - 4| = k$ has exactly one solution, where $k$ is a constant. What is the value of $k$?
Enter your answer.
Show solution
Answer: $0$. When $k > 0$ there are two solutions, $4 + k$ and $4 - k$. When $k < 0$ there are none. Only $k = 0$ gives exactly one solution, $x = 4$.

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