Desmos on the Digital SAT — Part 6. One specific calculator skill, shown on the real Desmos screen. Part of the full series; each part builds on the last.
Inequalities are where a lot of students freeze. A single line you can picture in your head. But "shade everything below it" — or worse, "which point satisfies both of these?" — is exactly the kind of question the built-in Desmos calculator is made to answer for you. On the Digital SAT, Desmos lives right inside Bluebook and stays available for the entire Math section — both modules — so this is a skill you can reach for again and again, not just once.
Type the inequality, watch it shade
The move is almost too simple. Click into an empty row and type the inequality the way it reads. For y <= 2x + 1, just type y <= 2x + 1. Desmos automatically converts <= into ≤ (and >= into ≥), so you never have to hunt for a special symbol on the keyboard.
The instant you finish typing, Desmos does more than draw the line y = 2x + 1. It shades every single point on the plane that makes the inequality true.
Now read what the shading is telling you. Because the inequality is y <= 2x + 1, Desmos shades the region below the boundary line — every point whose height (y) is less than or equal to the line's height at that x. The boundary line itself is drawn solid, which is Desmos's way of saying "points on the line count too" — that's the or equal to in <=. If you had typed a strict y < 2x + 1, that boundary would be dashed instead, to show those points are excluded.
That's the whole concept. The shaded region is the answer set. Every point inside it is a solution; every point outside it is not. You've turned an algebra question into a "is the dot in the shaded part?" question.
Test a point two ways — and let them agree
Here's the classic Digital SAT setup: "Which of the following is a solution to y <= 2x + 1?" You've got two independent ways to check, and Desmos lets you run both.
By hand: take the point (4, 0). Substitute the coordinates in: is 0 <= 2(4) + 1? That's 0 <= 9, which is true. So (4, 0) is a solution.
By graph: type (4, 0) into a fresh Desmos row. It drops a dot at x = 4, y = 0 — and you'll watch it land squarely inside the shaded region. No arithmetic required to see it.
When the two methods agree — the algebra says true and the dot sits in the shade — you can move on with real confidence. That agreement is your safety net against a careless sign flip.
Systems: plot every choice, find the overlap
Most SAT inequality questions don't stop at one line. They hand you a system — two inequalities at once — plus four answer-choice points. Something like:
"Which point (x, y) is a solution to the system y <= 2x + 1 and y >= -x + 3?"
Your plan:
- Type both inequalities on their own rows. Desmos shades each one, and where the two shadings pile on top of each other — the darker band where they overlap — is the solution region for the system. A point only counts if it's in that overlap.
- Type each answer choice as a point:
(4, 0), and the other three. - The correct answer is the one dot that lands inside the overlap. The dots that fall in only one shade, or in neither, are out.
You don't have to grind all four choices through substitution. Plot them, glance at which single dot is sitting in the overlapping region, and you're done. If two dots look uncomfortably close to a boundary, fall back to plugging those two into the inequalities by hand to break the tie — you've narrowed four suspects to two.
Where it shows up on test day
Inequality questions on the Digital SAT wear a few disguises. Sometimes it's a bare "which point is a solution" like the one above. Sometimes it's a word problem — a budget, a weight limit, a minimum number of hours — that you translate into <= and >= first, then graph. And sometimes the graph is already drawn for you and you're asked which inequality it represents; there, you reverse the trick — type a candidate inequality and see whether its shading matches the picture on your screen.
The common thread: once the inequality is typed into Desmos, the shading removes the guesswork. You're never asking "which side is below the line again?" — you're just looking at where the shading is.
One habit to build now: whenever you type an inequality, pause for half a second to confirm the right region shaded. A quick check with an easy test point — the origin (0, 0) is perfect when the line doesn't pass through it — tells you instantly whether you typed <= when you meant >=. That half-second has saved a lot of points.
Want to drill this until the shade-and-plot routine is automatic? UnlimitedTests has full-length Digital SAT Math sets in the same Bluebook-style Desmos environment. New here? Create a free account and run a timed module. Already signed in? Head to your dashboard and open a Math practice set — filter for systems of inequalities and run this exact workflow a dozen times.
Next in the series: Part 7 — statistics and regression straight from a data table.