The key idea
Translate “at most” as ≤ and “at least” as ≥, solve like an equation, then decide which whole numbers the context allows. Round in the direction the inequality forces, not to the nearest whole number.
The trap
Rounding 13.6 up to 14 when the limit says “at most.”
Can Desmos do it?
Graphing the inequality shows the boundary, but the whole-number decision at the end is yours.
College Board skill: Linear inequalities in one or two variables · Algebra
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
A club has $167 to spend on a party. The room costs $31, and each guest's meal costs $6. What is the greatest number of guests the club can feed without spending more than $167?
Show the answer
Answer: 2231+6g≤167 gives g≤6136≈22.67. The number of guests is a whole number, so the greatest is 22; rounding up would go over budget.
Level 2 · Test level
A van can carry at most 1049 kilograms. It carries a driver who weighs 73 kilograms, small boxes that weigh 22.5 kilograms each, and large boxes that weigh 40 kilograms each. If it carries exactly 3 times as many small boxes as large boxes, what is the greatest number of large boxes it can carry?
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Answer: 9With L large boxes there are 3L small ones: 73+22.5(3L)+40L≤1049, so 107.5L≤976 and L≤9.08. The greatest whole number is 9.
Level 3 · Beyond the test
Mira earns $15 per hour tutoring and $11 per hour at a café. This week she wants to earn at least $391 while working at most 28 hours in total. If she works only whole numbers of hours, what is the least number of hours she can spend tutoring?
Show the answer
Answer: 21To tutor as little as possible, fill the rest of the 28 hours at the café. With t hours tutoring: 15t+11(28−t)≥391, so 4t≥83 and t≥20.75. The least whole number is 21.
Practice all 36 questions
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