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SAT Systems of Inequalities and Shaded Regions

These questions show a graph with a shaded region, or give two inequalities and a list of ordered pairs. You either pick the system that produces the region or pick the point that lies inside it. Testing a point beats reasoning about the picture, and it works in both directions.

The rule

A point solves a system of inequalities only if it satisfies every inequality, so test it in all of them.

Start with the version that gives you a system and asks which ordered pair is a solution. A point is a solution only when it satisfies every inequality in the system, not just one. Take the system y is greater than 2x minus 3 and y is less than or equal to negative x plus 6, and test the point with coordinates 1 and 2. First inequality: 2 is greater than negative 1, true. Second: 2 is less than or equal to 5, true. Both hold, so the point works.

Reject as soon as one inequality fails. Testing the point with coordinates 4 and 7 in the same system: the first gives 7 is greater than 5, true, but the second gives 7 is less than or equal to 2, false. Stop there. Checking the remaining inequality wastes time on a point already eliminated.

Now the graph version. The boundary line tells you the equation, and the shading tells you the direction. Find the slope and intercept of the drawn line, which gives the equation, then pick any convenient point clearly inside the shaded region and substitute. If the shaded region includes the origin, use the origin, because substituting zero for both variables reduces the inequality to a comparison of constants.

The style of the boundary line encodes the equal-to part. A solid line means points on the line are included, so the inequality uses greater than or equal to, or less than or equal to. A dashed line means the boundary is excluded, so the inequality is strict. SAT answer choices often differ only in this detail, so read the line style before comparing anything else.

In context questions, the region is usually bounded by common sense as well as by the stated constraints. If x and y count physical objects or hours, both are at least zero, which quietly restricts you to the first quadrant. A word problem asking for the maximum number of one item will have its answer at a corner of that region, so check the corner points rather than scanning the interior.

The trap

Accepting a point because it satisfies the first inequality, without checking that it also satisfies the second.

Try it

Try one
Real question

Solve -(x - 5) > 2.

A greenhouse must keep its relative humidity above 58 percent and at or below 76 percent. The humidity, in percent, is modeled by 34 + 3m, where m is the number of misting cycles run per hour. What is the greatest number of misting cycles per hour allowed?

  • A) 6
  • B) 8
  • C) 14 ✓
  • D) 15

Why: Set 34 + 3m <= 76: 3m <= 42, so m <= 14. Answering 6 means you divided the range between the two limits by 3. Answering 8 means you used the 58 percent lower limit instead of the upper limit. Answering 15 means you took one more cycle, which puts the humidity at 79 percent, above the limit.

Practice more of these free

Questions students ask

What does a dashed boundary line mean?

It means the points on the line itself are not part of the solution, so the inequality is strict, using greater than or less than. A solid line means the boundary points are included, so the inequality uses greater than or equal to or less than or equal to.

Which test point should I use?

Use the origin whenever it is not on the boundary line, because substituting zero for both variables makes the arithmetic almost disappear. If the origin sits on the line, pick any other point clearly inside the region you are checking.

Guides that cover this

More on linear inequalities