Linear Inequalities
An inequality describes a range of values, not a single point, and solving one means finding every number that satisfies it. The algebra mirrors solving an equation, with one rule that has no equation analog: multiply or divide both sides by a negative and the symbol has to flip. The points leak there, and in the boundary: whether a strict or inclusive symbol keeps the endpoint, and whether the answer is a value in the set or the edge sitting just outside it.
These patterns aren't really about whether you can isolate a variable. They're about whether you flipped the symbol when the sign demanded it, whether you read each endpoint the way its symbol and its dot dictate, and whether you reported the exact value the item asked for, the greatest integer in the set, not the boundary just outside it.
The work
4 ways in · any order
Lesson
Linear Inequalities
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Solve a linear inequality and read its solution as a range, flip the symbol when you multiply or divide by a negative, tell a strict end from an inclusive one on a number line, and handle a compound two-sided range. The lesson works the method and the three patterns that derail it, and it saves the trap for last: reporting the boundary when the item asked for a value inside the set.
Diagnostic
10-item topic check
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Ten items across the three patterns: forgetting to flip the symbol on a negative, mixing up strict and inclusive endpoints, and reporting the boundary when a value in the set was asked. A mix of solve-and-report, boundary reads from a number line, compound ranges, and a feasible region. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.