SAT Math · Heart of Algebra
Linear inequalities and systems of inequalities
Solving inequalities and reading shaded regions in the plane.
What's tested in this subtopic
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).
Tactics that actually move your score
When solving, treat the inequality like an equation but watch for the sign-flip when you multiply or divide by a negative. For systems with a feasible region, plug each candidate point into each inequality individually rather than trying to graph mentally. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy A student has a budget of $48. Each notebook costs $3 and a backpack costs $3. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Easy A student has a budget of $26. Each notebook costs $5 and a backpack costs $4. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Medium A student has a budget of $58. Each notebook costs $7 and a backpack costs $5. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Medium A student has a budget of $47. Each notebook costs $3 and a backpack costs $6. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Medium A student has a budget of $26. Each notebook costs $7 and a backpack costs $18. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Hard A student has a budget of $29. Each notebook costs $3 and a backpack costs $18. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Easy 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…
- Easy A student has a budget of $53. Each notebook costs $3 and a backpack costs $7. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Medium A student has a budget of $21. Each notebook costs $3 and a backpack costs $19. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Medium A student has a budget of $24. Each notebook costs $6 and a backpack costs $18. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Medium A student has a budget of $42. Each notebook costs $5 and a backpack costs $6. If she buys exactly one backpack and x notebooks, what is the maximum value of x…
- Hard A student has a budget of $37. Each notebook costs $2 and a backpack costs $10. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Easy A student has a budget of $52. Each notebook costs $5 and a backpack costs $11. If she buys exactly one backpack and x notebooks, what is the maximum value of …
- Easy A student has a budget of $49. Each notebook costs $7 and a backpack costs $10. If she buys exactly one backpack and x notebooks, what is the maximum value of …
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in SAT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.