Desmos for the SAT · Technique 19 of 23

Graph a System of Inequalities to Test a Point

Shaded regions overlap exactly where every inequality agrees.

SolvesSystems of inequalitiesTest a pointWhich ordered pair satisfies the system?

Watch the lesson

The whole move worked on one real SAT question, narrated.

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

Desmos shades the solution region for any inequality the moment you type it in with <, >, <=, or >=. Type < then = to get <=, and > then = to get >=.

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.

To test whether a point satisfies the system, type its coordinates, like $(3,4)$ , on its own line and check the Label box next to it. Desmos plots the point on the graph, so you can see which shaded regions it falls in.

One inequality in x alone is a different picture, a shaded band with one edge: that is solving an inequality by graphing. Four inequalities describing a word problem is which inequality represents this situation.

Nova

A system of inequalities usually gives you four ordered pairs and asks 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 the one sitting in the overlap.

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 substituting each point into each inequality by hand.

The question

$y\ge-x+4$ and $y<3x-2$

Which of $(3,4)$ , $(2,4)$ , $(1,5)$ and $(0,0)$ satisfies both?

The algebra way

Take a choice and substitute it into y >= -x + 4. For (3, 4): 4 >= -3 + 4, so 4 >= 1 holds.
Substitute the same point into y < 3x - 2. For (3, 4): 4 < 9 - 2, so 4 < 7 holds.
Both held, so (3, 4) works. Any choice that fails either check is out, and a point that lands exactly on a strict boundary, like (2, 4) on y = 3x - 2, fails that inequality.

The Desmos way

Type y >= -x + 4 and y < 3x - 2 so Desmos shades each region.
Plot the four choices with Label checked. (3, 4) is the only dot inside the overlap, and (2, 4) sits on the dashed edge, so it is out.

The payoff: One picture replaces eight substitutions, and the only arithmetic left is for the one dot that lands on an edge.

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

The whole move, all 4 steps

Type the first inequality on its own line, using <=, >=, < or > exactly as written.
Enter every other inequality in the system, one per line.
Find the region where all the shaded areas overlap. That overlap is the solution set.
Plot the points the question offers, with Label checked, and see which one lands inside the overlap.

The question

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

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

A $(3, 4)$
B $(2, 4)$
C $(1, 5)$
D $(0, 0)$

Step 1 of 4

Type the first inequality exactly as printed: y >= -x + 4. Type > then = for the $\ge$ symbol. Desmos shades everything above a solid line.

In the calculator: y>=-x+4

Now try it on a real SAT question

Head or scratch paper. Nothing to graph. This asks what a term means, not where a region is. Read the sentence and match the parts.

Anthony will spend at most $115 to buy x small cheese pizzas and y large cheese pizzas for a team dinner. The inequality $11x + 14y <= 115$ represents this situation.

Which of the following is the best interpretation of 14y in this context?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

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.

The Desmos trap: when to close the calculator

If the question hands you one point and asks yes or no, substitute by hand. Two substitutions are faster than opening the calculator and shading. Graphing is the move when you are choosing among several candidate points.
A strict inequality (< or >) draws a dashed edge, and the line itself is not part of the solution. A point that looks like it sits on a dashed edge is the trap the question was built around, so settle it with arithmetic: evaluate the boundary at the point's x and compare, the way $3(2)-2=4$ settles $(2,4)$ .
A <= or >= inequality draws a solid edge that is included, so a point exactly on a solid boundary of the overlap is a solution. The boundary table on this page has the four cases.

All Desmos techniques

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.