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700-800 level patterns, traps, and speed techniques.
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At this scoring band, linear questions are no longer about solving โ they are about structure recognition and answering the exact quantity asked. Three archetypes dominate the hard tier.
Parameter equations. Given , no solution means the x-coefficients match but constants differ; infinitely many means both match. The coefficient equation hands you the candidate in one step; the constants decide which case you are in.
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Solve-then-answer-something-else. Hard word problems make you find x, then ask for the total, the difference, or an expression like (x + 3)/2. Your correctly computed intermediate value is always planted in the options.
Constrained maximums. Budget and capacity items layer flat fees, per-unit rates, and conditional surcharges past a threshold, then punish rounding: budgets round down, "at least" requirements round up, and strict "less than" comparisons exclude the break-even value itself. Always verify the boundary integer by direct arithmetic.
The distractor molds are equally predictable: the intermediate value, the boundary value, the wrong rounding direction, and the sign-flip casualty from dividing an inequality by a negative. Reading the final sentence first and re-checking which quantity is requested converts these from traps into free eliminations.