Linear Inequality Systems

When one constraint isn't enough — how to combine two inequalities and find the region where both conditions are met.

You've already worked with systems of equations — two equations, one intersection point. And you've worked with single inequalities — one constraint, a range of values that satisfy it. A system of inequalities puts both ideas together. Instead of finding one point where two lines cross, you're finding the entire region where two shaded areas overlap.

That overlap is your answer. Every point inside it satisfies both conditions at the same time. The SAT tests this idea in three different ways, and each one asks you to do something slightly different with the system. This article breaks down all three.

What makes these different

With a system of equations, each equation is a line, and the solution is the single point where those lines intersect. One point, one answer.

With a system of inequalities, each inequality is a line plus a shaded region — everything on one side of the line. When you combine two inequalities, you get two shaded regions, and the solution is where they overlap. That's not a point — it's an area. Every coordinate pair inside that area is a valid solution.

In Desmos, this is easy to see. Type both inequalities in and each one shades a region. Where the two shaded regions overlap — the area that's darker or where both colors blend — that's where both conditions are met. Any point in the overlap is valid. Any point outside it fails at least one condition.

How the SAT tests this

There are many ways a system of inequalities can show up on the SAT, but three formats come up far more often than the rest:

• Setup questions — a word problem with two constraints, and you're asked which system of inequalities represents the situation. Pure translation, no solving.
• Point-checking questions — a system is given, and you're asked which ordered pair satisfies both inequalities. You're substituting and testing, not setting up.
• Max/min questions — a word problem with two constraints, and you're asked for the maximum or minimum value of one variable that satisfies both. This is the hardest type — you set up the system, figure out which constraint to push to its limit, and solve.

Each one requires a slightly different skill, so this article walks through them one at a time with examples.

Setup questions

The most common form on the SAT. You get a word problem with two constraints and the question asks: which system of inequalities represents this situation? You're not solving anything — you're translating words into math, one constraint at a time.

The process is the same as setting up a single inequality — you just do it twice. Read each constraint separately, translate it into its own inequality, and combine. The key is keeping the two constraints separate in your head. Each fact in the problem gives you one inequality. Don't try to mash them into a single expression — build each one on its own.

Point-checking questions

The second form gives you a system of inequalities and a list of ordered pairs. The question asks: which point satisfies both inequalities?

The fastest approach is Desmos. Graph both inequalities — Desmos shades each one, and the overlap region is where both conditions are met. Then type each answer choice as a point directly into Desmos. If a point lands inside the overlap, it satisfies both inequalities. If it lands outside, it fails at least one. You don't need to do any arithmetic — just graph and look.

Max/min questions

The hardest form. A word problem gives you two constraints and asks for the maximum or minimum value of one variable that satisfies both.

The first step is the same as a setup question — read the constraints, translate each one into an inequality. The difference is what comes next. Instead of picking the right system from a list, you graph both inequalities in Desmos and look at the overlap region. The maximum or minimum value of the variable you need is at the edge of that region — specifically at the corner where the two boundary lines meet. Click that corner point in Desmos and read the value.

Quick practice

Try these on your own. Pick an answer and you'll see right away whether it's correct, along with the reasoning.