Forbidden values
SAT glossary · Course Foundations
Definition
The inputs that make an original denominator zero, written down the instant you see the denominator and BEFORE solving. In (x)/(x-3) = (3)/(x-3)+2 the flawless algebra yields x=3 — the forbidden value — so the equation has no solution. The purest example on the exam of correct work producing a wrong answer, and Desmos is actively misleading here because it draws over the hole.
Forbidden values in plain English
'Forbidden values' are the specific input numbers that would make an ORIGINAL denominator (before you do any algebra) equal to zero — and you should identify these the instant you see the denominator, before you even start solving the equation. Here's why that matters: in an equation like (x)/(x-3) = (3)/(x-3)+2, doing the algebra perfectly and correctly can still lead you to the answer x=3 — except x=3 is exactly the forbidden value that breaks the original denominators, meaning the equation actually has NO solution at all, even though your algebra never made a mistake. This is one of the rare cases where doing everything right still gives you a wrong final answer if you forget to check it against the forbidden value — and graphing tools like Desmos can trick you here too, since they sometimes draw right over the gap (hole) without showing it.
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