When the answer isn't one number — how to set up and solve word problems that deal with ranges, limits, and constraints.
If you've already worked through linear equation word problems, inequality word problems will feel familiar. The setup process is almost identical — you're still identifying a rate, a starting value, and building an expression. The difference is that instead of finding one exact answer, you're finding a range of values that work.
That shift is small on paper, but it changes how you read the problem, how you write the math, and how you interpret your answer.
What makes these different
In a linear equation word problem, the question usually asks you to find a specific value. “How many hours did she work?” “What was the total cost?” There's one answer.
In an inequality word problem, the question asks about a constraint. “How many hours does she need to work to earn at least $500?” “What's the maximum number of items he can buy without going over his budget?” The answer isn't a single number — it's every number that satisfies the condition.
Any time a problem uses words like “at least,” “no more than,” “at most,” “fewer than,” “more than,” or “must exceed,” you're dealing with an inequality.
Translating the language into math
This is the part that trips students up the most. You know it's an inequality, but you're not sure which direction the sign should face. Here's a reference:
| Phrase | Symbol | What it means |
|---|---|---|
| “at least” | ≥ | that amount or more |
| “no more than” | ≤ | that amount or less |
| “at most” | ≤ | that amount or less |
| “more than” | > | strictly above, not equal |
| “fewer than” / “less than” | < | strictly below, not equal |
| “must exceed” | > | strictly above, not equal |
| “no fewer than” | ≥ | that amount or more |
The distinction between ≥ and > (or ≤ and <) matters. “At least 10” means 10 counts — it's ≥. “More than 10” means 10 doesn't count — it's >. The SAT is precise about this, so you should be too.
After you write the inequality, test it with the boundary number. If the problem says “at least 500” and you wrote ≥ 500, ask yourself — does exactly 500 satisfy the condition? Yes. If you wrote > 500, then 500 wouldn't count, which contradicts “at least.” That one-second check can save you from picking the wrong answer.
Setting up and solving
The actual process is the same as linear equations. Find the fixed amount, find the rate, identify your variable, and build the expression. The only difference is that you place an inequality sign instead of an equals sign.
This rule doesn't come up in every problem, but when it does, forgetting it gives you the opposite of the correct answer. For example, if you have −3x > 12 and you divide both sides by −3, it becomes x < −4, not x > −4.
Desmos is a useful tool for checking your work on inequality problems. Type an inequality directly into Desmos and it shades the valid region — every value in the shaded area satisfies the condition. This gives you a quick visual confirmation that your algebra is right.
Worked examples
Quick practice
Try these on your own. Pick an answer and you'll see right away whether it's correct, along with the reasoning.
The process is the same every time: read the problem, identify the constraint language, build the expression, place the right inequality sign, and interpret. Graph the inequality in Desmos to confirm — the shaded region shows you every value that works, and the boundary tells you whether equality counts. Pay close attention to that boundary. It's where the SAT likes to test you.