Rational, Radical, and Absolute-Value Equations

These three equation types share a habit the SAT keeps testing: the answer is not whatever the algebra produces, it is whatever survives a check. The points leak in three moves. A radical equation gets squared, a value appears, and it is handed in without being tested, even though squaring can introduce an extraneous solution that the original equation rejects. A rational equation gets cleared and solved, and the value reported is exactly the one a domain restriction forbids, because it makes a denominator zero. And an absolute-value equation gets solved as a single case when it opens into two, or solved at all when a negative right side means it has no solution.

None of this is really about whether you can solve the equation. The algebra is short and you can almost certainly do it. The points turn on what you do once you have a candidate: substitute it back into the original to catch an extraneous root, read the denominators before you trust a value, and open both cases of an absolute value while checking that the right side even allows one. Solve, then check, then report only what holds.