Desmos for the SAT · Technique 18 of 23
Which Inequality Represents This Situation?
The words pick the symbol. The shading checks the rest.
Watch the lesson
The whole move worked on one real SAT question, narrated.
A short story, a limit, and four inequalities underneath. It looks like the which equation represents this situation question you have already worked through, and the method is a cousin of it, but the thing being tested is different. An equation has one job to get right: what sits on each side. An inequality has three, because it also has a direction and an edge, and the direction and the edge are where these questions are decided.
The direction comes from the sentence, and there is no route around reading it. At least and no less than mean the value may sit on the limit or above it, so >=. More than, over and exceeds mean past the limit, so >. At most, no more than and a maximum of mean the limit or below, so <=. Less than, under and below mean short of the limit, so <. A fair number of these questions are finished right there, before the calculator is even open.
Everything after that, Desmos can show you. Type the situation as an inequality in the letters the question uses, and Desmos shades every pair of values that satisfies it. Type a choice on the line underneath and its shading lands on top. A choice with the same solution set covers the first shading exactly, so none of the first color is left showing. A wrong one gives itself away in one of three shapes: it shades the other side of the edge, it sticks out past the edge in a strip, or its edge tilts at a different angle. This is a check on your translation, not a replacement for it. The situation you type has to come from the sentence, and when the choices differ only by the symbol there is nothing to type.
Two neighbors of this question look similar and take different moves. One inequality in x, asking for its solution set, is solving an inequality by graphing. Two inequalities and four ordered pairs is graphing a system of inequalities.
Nova
Two things decide this question, and only one of them is math. The words set the symbol, and no calculator reads them for you. Once that is settled, type the situation as an inequality and put each choice under it. Desmos shades both, and a choice that describes the same van covers the first shading exactly. One that does not shows you its mistake as a shape.
Now you try it
The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.
The algebra way vs the Desmos way
This technique sidesteps rearranging the sentence into the form the choices are written in.
The question
A van holds at most 4,200 pounds, is already carrying a 350-pound refrigerator, and the rest of the load is x boxes at 18 pounds each and y boxes at 12 pounds each.
Which of the four printed inequalities represents this?
The algebra way
- Take the refrigerator out of the limit first:
4200 - 350 = 3850, so the boxes have 3,850 pounds between them. - Attach each weight to the right count: 18 pounds per box of books is
18x, and 12 pounds per box of clothing is12y. - Read the limit phrase. A maximum is a ceiling the load may reach, so the symbol is
<=, not<and not>=. - Write
18x + 12y <= 3850. Three separate decisions, and nothing on the page checks any of them.
The Desmos way
- Type the sentence the literal way, with no rearranging:
350+18x+12y<=4200. Desmos shades every load the van can take. - Type the choice you believe on the line underneath.
- If its shading covers the first one exactly, it fits. If it shades the other side, sticks out in a strip, or tilts, it is describing a different van.
The payoff: The algebra asks for three decisions and gives you nothing to check any of them against. The graph checks all three at once: a choice that rearranged the sentence correctly has the same picture as the sentence, and one that did not shows you which decision went wrong.
Quick reference
The phrase decides the symbol, and whether the limit itself is allowed
| The question says | Symbol | Can the value sit exactly on the limit? |
|---|---|---|
| at least, no less than, a minimum of, not below | >= | Yes |
| more than, over, exceeds, greater than | > | No, it has to be past the limit |
| at most, no more than, a maximum of, up to, within | <= | Yes |
| less than, under, below, fewer than | < | No, it has to stay short of the limit |
| between a and b, from a to b, varied between a and b | a <= v <= b | Yes at both ends |
What the second shading tells you when a choice lands on the situation
| What you see | What it means |
|---|---|
| It covers the first shading exactly, and none of the first color is left | Same solution set. The choice fits. |
| It shades the other side of the same edge | The symbol faces the wrong way |
| It covers the first shading and sticks out past the edge in a strip | The choice allows more than the situation does: a fixed amount was never taken out of the limit, or the number on the right is too big |
| It stops short of the edge and leaves a strip of the first color showing | The choice allows less than the situation does: a number is too small, or a repeating amount is charged too often |
| Its edge tilts at a different angle and crosses the first edge | The two rates are attached to the wrong letters |
| Same region, but one edge is dashed and the other solid | Strict against non-strict. The phrase settles it, or {limit<=limit:1,0} against {limit<limit:1,0} |
Step by step
The whole move, all 7 steps
- Start with the phrase that sets the limit and read the symbol straight off it. The phrase table on this page covers the direction and whether the limit itself is allowed.
- Check what the four choices actually disagree about. If they share a left side and differ only in the symbol, the words have already answered the question and there is nothing to type.
- Type the situation as an inequality in the letters the question uses, with nothing carried from one side to the other. Desmos shades every pair of values that satisfies it, and it only shades in x and y, so a question written in other letters is typed in those two.
- Type the choice you believe on the line underneath. If it describes the same situation, its shading covers the first one exactly and none of the first color is left showing.
- Read a wrong choice by its shape. Shading on the other side of the edge is a symbol facing the wrong way. A strip past the edge, or a strip of the first color left uncovered, is a wrong number on one side. An edge at a different angle is two amounts attached to the wrong letters.
- Hide a choice once you have read it, by clicking the colored icon at the left of its row, so the next one lands on the situation alone.
- If two choices shade the same region and differ only in
<=against<, look at the edge: a strict symbol draws it dashed and a non-strict one draws it solid. To see the difference as a number, type the limit against itself:{limit<=limit:1,0}returns one and{limit<limit:1,0}returns zero.
The question
A delivery van can carry a maximum weight of 4,200 pounds. During one trip, the van will be used to carry a 350-pound refrigerator as well as several boxes of books weighing 18 pounds each and several boxes of clothing weighing 12 pounds each.
If x represents the number of boxes of books and y represents the number of boxes of clothing, which inequality best represents this situation?
- A
- B
- C
- D
Step 1 of 6
Read the limit phrase before touching the calculator. A maximum weight of 4,200 pounds is a ceiling the van is allowed to reach, so the symbol is . That alone rules out choice B, which points the other way.
Desmos does not read English, so this decision is yours either way. Making it first also means one less line to type.
If you typed B anyway, it would shade the other side of the same edge. The two colors meet at the boundary and neither covers the other, which is what a symbol facing the wrong way looks like.
Now try it on a real SAT question
Head or scratch paper. All four choices carry the same left side. Nothing is in question except the symbol, and the calculator cannot read the phrase that sets it. At most means the company is allowed to spend the budget right down to the last dollar, so the limit stays in and the symbol is . Ten seconds of reading.
A company has a budget of at most 5,000 dollars for advertising. Radio ads cost 150 dollars each and online ads cost 75 dollars each.
If r represents the number of radio ads and n represents the number of online ads, which inequality represents this situation?
Pick an answer to get instant coaching from Nova.
Every choice carries the same left side, so hold off on typing. Read the phrase that sets the budget and ask one thing: is the company allowed to spend the limit exactly? If you want that check on screen, type and see what comes back.
Each radio ad costs 150 dollars and each online ad costs 75 dollars, so the total spend is , which is the left side of all four choices. At most 5,000 dollars means the total may be anything up to and including the budget, so the symbol is .
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
Set this question up yourself, then read the answer straight off the graph.
Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.
Straight from Desmos
- Desmos Help Center: InequalitiesHow Desmos shades inequalities, and what the dashed and solid boundaries mean.
The Desmos trap: when to close the calculator
- Desmos cannot read the sentence. The symbol comes from the words, and when the four choices share a left side and differ only in
<,<=,>and>=, the calculator has nothing to add. - This is a check on your translation, not a substitute for it. The situation you type has to come from the sentence, with nothing carried across, or you are checking the choices against a guess.
- Desmos shades in x and y only. A question written in r and n, or in m, gets a slider offer instead of a shading until you type it in x and y.
- Hide a choice after you have read it. Four shadings stacked on one situation blend into each other; one at a time, each shape is plain.
- Identical shading, different edges. A strict symbol draws its edge dashed and a non-strict one solid, and that is the only visible difference between
<and<=. When no whole-number load lands exactly on the limit, as in the advertising question, the phrase is the only evidence.
If the calculator does something you did not expect
- Zoom out until the edge is on screen. In the default window the van question shades the whole screen for every choice except B, and four full screens of color look identical.
Nova
Desmos does not read English. When the four choices share a left side and differ only in the symbol, the answer lives in the phrase that set the limit, and there is nothing worth typing. Open the calculator on the ones where the choices also disagree about what belongs on each side, and type the situation from the sentence, not from a choice.
Put this move into real practice
"Which Inequality Represents This Situation?" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.
Practice freeSAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.