Absolute value equations
An absolute value equation looks small but hides a second answer. Reading the bars as distance explains why, why a negative target is impossible, and why some solved answers have to be thrown out. Get the split right and the rest is ordinary algebra.
Absolute value as distance
The equation |x - 3| = 5 says the distance between x and 3 is 5. So x sits five steps to the right of 3 or five steps to the left, giving 8 or -2. Reading it as distance explains why there are usually two answers and why an absolute value can never equal a negative number.
The two case split
Write the inside equal to the positive value and equal to its negative. For |2x + 1| = 9, solve 2x + 1 = 9 and 2x + 1 = -9. That gives x = 4 and x = -5. Always split, because skipping the negative case quietly loses half the answer and the question is built to reward catching it.
Why some solutions get rejected
When the variable sits on both sides, an answer can fail. In |x + 2| = 3x - 1, the right side must be positive for any solution to exist. Splitting gives x = 3/2 and x = -1/4, but the second makes the right side negative, which is impossible. Only x = 3/2 survives.
Check by substitution
Always plug both answers back into the original equation. The arithmetic takes seconds and it catches the case that looks fine but fails. If an answer would make an absolute value equal a negative number, it cannot be correct, no matter how clean the algebra looked on the way there.
Why do absolute value equations have two answers?
Because the distance inside the bars can come from a positive value or its negative. Both sit the same distance from the centre.
Can an absolute value equal a negative number?
No. Distance is never negative, so if an equation asks for a negative result there is no solution.
When do I reject a solution?
When it makes one side of the original equation impossible, usually by forcing an absolute value to equal a negative. Substitute to check.
What if the equation equals zero?
There is only one answer. The inside must be zero, and the negative case gives the same value, so the two cases collapse.
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Updated 2026-09-26. Free forever, no account needed.