Desmos for the SAT · Technique 11 of 15

Graph a System of Inequalities to Test a Point

Shaded regions overlap exactly where every inequality agrees.

SolvesSystems of inequalitiesTest a pointFeasible regions

Desmos shades the solution region for any inequality automatically the moment you type it in with <, >, <=, or >=. Two shortcuts save time here: type < then = to get <=, and use | or abs for absolute value.

For a system of two or more inequalities, enter each one on its own line. The region where all the shading overlaps is the full solution set, and any point in that overlap satisfies every inequality at once.

Think of it as the darkest region. The darkest double-shaded overlap is the only solution set, so any point that lands outside it is a trap answer, no matter how close it sits to the shading.

To test whether a point satisfies the system, type its coordinate like (4,5) on its own line and check the Label box next to it. Desmos overlays the point right on the graph, so you can see exactly which shaded regions it falls in.

Nova

Systems of inequalities love to hand you four ordered pairs and ask which one fits both rules. Instead of plugging every point into every inequality, you type the two lines once, let Desmos shade, and the right answer is just chilling in the overlap.

The algebra way vs the Desmos way

This technique sidesteps testing points in each inequality by hand.

The algebra way

Take a candidate point and substitute it into y <= -2x + 6. For (0, 0): 0 <= -2(0) + 6, so 0 <= 6 holds.
Substitute the same point into y > x - 1. For (0, 0): 0 > 0 - 1, so 0 > -1 holds.
Both held, so (0, 0) works. If a point fails either check, repeat the substitution for the next candidate.

The Desmos way

Type y <= -2x + 6 and y > x - 1 so Desmos shades each region.
See which candidate point lands in the darker overlap. (0, 0) sits inside both, so it satisfies both.

The payoff: One glance at the overlap replaces two substitutions and two comparisons per candidate point.

See the move, live

That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.

Desmos graphing calculator, livepre-loaded

Quick reference

Is a point on the boundary a solution?

Where the point sits	Solution?
Inside the darkest overlap	Yes
On a solid boundary of the overlap	Yes
On a dashed boundary	No
Outside the overlap	No

Step by step

Nova

Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.

Step 1 of 4

Type each inequality exactly as given, e.g. y <= -2x + 6.

Desmos showing the first inequality y <= -2x + 6 with its solution region shaded.

Now try it on a real SAT question

A system of inequalities is given: y >= -x + 4 and y < 3x - 2.

Which ordered pair (x, y) satisfies both inequalities?

Pick an answer to get instant coaching from Nova.

Nova

Graph both, look for the darker overlapping patch, and any point sitting inside it works for both inequalities at once.

Your turn, blank calculatornothing loaded

Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.

Watch it

Type both inequalities, let Desmos shade, and read the overlap.

The Desmos trap: when to close the calculator

If you only need to check whether one specific point satisfies a written inequality, plugging the numbers in by hand is faster than graphing.
For a strict inequality (< or >), remember the boundary line itself is never part of the solution. If a candidate point lands exactly on the line, work it out algebraically instead of squinting at a dashed line on screen.
Watch the solid-versus-dashed boundary. A strict inequality (< or >) draws a dashed boundary, and Desmos does not shade the line itself, so a point sitting exactly on a dashed line is not a solution. A <= or >= inequality draws a solid boundary that is included. If a point looks like it sits on a boundary, hover over the line to check whether it is dashed before you trust the shading.

Nova

Real talk though: if the question already hands you one point and asks yes or no, just plug the numbers in. Two quick substitutions beat opening the calculator and shading regions. Graphing is your move when you're picking between a bunch of candidate points, not for a single yes-or-no check.

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Put this move into real practice

"Graph a System of Inequalities to Test a Point" 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 free

SAT 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.