Linear Equations & Inequalities at the 700-800 Level

At this scoring band, linear questions are no longer about solving — they are about structure recognition and answering the exact quantity asked. Three archetypes dominate the hard tier.

Parameter equations. Given $a(2x - 1) + 7 = 6x + 3$, 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.

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.