SAT · Heart of Algebra · Linear inequalities and systems of inequalities
Question sat-q-00049
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The question
A student has a budget of $41. Each notebook costs $5 and a backpack costs $9. If she buys exactly one backpack and x notebooks, what is the maximum value of x she can afford?
- 6
- 5
- 7
- 32
Show answer & worked solution
Answer: 6
Worked solution. Set up the inequality 5x + 9 ≤ 41. Subtract 9: 5x ≤ 32. Divide by 5: x ≤ 6.4. The maximum integer x is 6.
Why each wrong choice is wrong. Off-by-one distractors catch students who solve correctly but round up instead of taking the floor. The fourth distractor is the leftover dollars after the backpack — it ignores the per-notebook division.
Test-day tactic. Whenever a real-world inequality has an integer answer, take the floor — buying 7.4 notebooks doesn't make sense.
About this question type
A linear inequality is solved like a linear equation, except multiplying or dividing both sides by a negative reverses the inequality. Systems of linear inequalities define a feasible region in the plane; the SAT often asks which (x, y) point satisfies all inequalities, or for the maximum value of a linear expression on a feasible region (introductory linear programming, without using that name).
You will see a question shaped like this one on roughly every other official SAT form, typically at an early position in the section, where missing them carries an outsize penalty because the curve assumes everyone in your band gets them right. Treat any miss in this subtopic as a signal to drill the subtopic page before you do another full practice test.
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