Hard Algebra

• Work top-quartile algebra questions one at a time at test pace, about 105 seconds each
• Apply backsolving and plug-in-numbers strategies on problems where direct algebra is slow
• Recognize common hard-problem setups (systems with parameters, word problems with multiple unknowns, compound inequalities)
• Build comfort with problem shapes that typically appear in the last 10 questions of ACT Math
• Identify which hard-problem archetypes you still miss most often

This is a drill day, not a teaching day. The assumption is that you have already studied algebra fundamentals. Today is about applying them under pressure to questions that look intimidating. Hard ACT algebra problems tend to share a few shapes: (1) Systems with a parameter. 'For what value of k do the equations 2x + 3y = 6 and 4x + ky = 10 have no solution?' The trick is recognizing that parallel lines need equal slopes and different intercepts, so k = 6. (2) Word problems with two or more unknowns where you must infer the second equation from context. (3) Inequality word problems involving 'at least' or 'at most' combined with a mixture or rate setup. (4) Rational equations that require clearing denominators and checking for extraneous solutions. (5) Function composition with an unknown on the inside: 'If f(x) = 2x + 1 and f(g(x)) = 6x − 3, find g(x).' Today's strategies: if a problem has answer choices that are small integers, backsolving is almost always faster than solving directly. If a problem has variables in the answer choices, picking numbers (substitute x = 2) and testing is usually faster. If the algebra path takes more than 90 seconds, switch strategies. Keep a notes column beside your scratch work: for each missed question, write the trap type (did you misread? use the wrong formula? run out of time?). Patterns in your miss log tell you what to drill next. After the drill, spend as much time reviewing as you spent taking it. Review is where the learning happens.

• Give each question about 105 seconds. If one runs over, guess, mark it, and move on.
• Backsolve whenever answer choices are small integers. Picking numbers works whenever answers have variables.
• If no plan comes to mind within about 30 seconds of reading a problem, mark it, skip it, and come back.
• Review missed questions the same day, not tomorrow. Log the error type for each miss.
• A hard drill is not a score test. A 50% raw score on top-quartile questions is strong. Do not panic if this feels brutal.