SAT Math — inequalities, boundaries and rounding

An inequality is solved almost exactly like an equation, which is what makes it dangerous. The arithmetic feels familiar, and the three things that are genuinely different about it are the three things a learner forgets under time pressure: whether the boundary value itself belongs to the answer, whether the direction has to turn around, and which way a decimal answer rounds when the situation rather than the arithmetic decides.

Two questions here are pure translation, one on a phrase meaning a ceiling and one on a phrase meaning a floor. In both, two of the four options are the strict version of the same relation, so the question cannot be answered by noticing which way the sign points — a learner has to decide whether spending exactly the whole budget, or exercising exactly the stated minimum, counts. It does, and the explanations show the concrete case that proves it.

Two are solving, and one of them divides by a negative number so that the direction reverses. That single step accounts for more lost marks on this topic than anything else, and the explanation does not merely state the rule: it substitutes a value from the unreversed answer and shows the inequality failing.

Two are compound, of the kind with an expression trapped between two numbers, where every operation must be applied to all three parts and in the right order. One asks for the range, one for the number of integers inside it — a count that turns entirely on whether the endpoints are included, with the wrong counts built from including one, both, or neither.

Two are questions where a budget or a capacity produces a decimal and the answer must be a whole number. One of them rounds down from a value whose decimal part is large, which is exactly where instinct fails: the next whole item costs more than the budget allows, so the arithmetic's preference is irrelevant. The other reaches its limit exactly and needs no rounding at all, which is a different trap. Both have a wrong option built from ignoring the one-off fee, the commonest setup error in this kind of problem.

Two present a pair of conditions at once — a spending cap together with a minimum quantity — and ask which pair of values clears both. The wrong options are built so that each fails a different condition, and one of them fails narrowly, because a pair that comfortably passes the test you happen to check first is precisely the one that catches people out.

The last two ask only whether a given number satisfies an inequality, including one worded in the negative, where solving correctly and then picking a value that works is a way of answering a question nobody asked.

Every sum of money is written in words, every store and club is invented, and nothing is drawn from any official publication.

• Translate "at most" and "at least" into inequalities, deciding deliberately whether the boundary value is allowed
• Reverse the direction after multiplying or dividing by a negative number, and verify by substituting a value
• Work on all three parts of a compound inequality, in the right order, and count the integers inside it correctly
• Round a decimal answer in the direction the situation requires rather than the direction arithmetic suggests
• Test a candidate pair against both conditions of a system, not only the one checked first
• Answer a question worded in the negative without solving it correctly and then picking a value that works

Written for this catalogue, with no source document. The content is the inequalities material named in the College Board's own public description of the Algebra domain — linear inequalities in one and two variables and systems of them; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test also asks some questions in a student-produced response format, which a multiple-choice bank cannot reproduce. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.