Interpreting a Point in the xy-Plane as a Solution
A point is a solution to a system of inequalities only if plugging its x and y values into every inequality makes each one true. One failure sinks the whole point.
A single inequality like y > x - 2 splits the plane into two halves. Every point on one side makes the statement true; every point on the other side makes it false. Instead of imagining the whole shaded region, you can just test one point: plug its x into the right side, compare to its y, and see if the inequality holds. That one check tells you everything about where that point sits, because the boundary line is the only place truth can flip.
A system asks for more: the point has to satisfy every inequality at once. So you test the point against each inequality separately, in any order, and the point only counts as a solution if it passes all of them. Fail even one, and the point is out, no matter how well it does on the others. This is exactly why systems shade as an overlap of regions on a graph: the solution set is wherever all the shadings stack.
This idea connects directly to graphing systems of inequalities and to solving systems algebraically. Testing a point is the fast, no-graph-needed version of the same question a picture answers visually: is this (x, y) in the overlap? It also sets up why boundary lines matter so much, since whether the line is solid or dashed decides if points sitting exactly on it belong to the solution set.
Key terms
- Solution set
- All the points (x, y) that make every inequality or equation in a system true at once.
- Boundary line
- The line where an inequality turns into an equation, separating the plane into a true side and a false side.
- Strict inequality
- An inequality using < or >, drawn as a dashed line, that excludes points sitting exactly on the boundary.
- Inclusive inequality
- An inequality using ≤ or ≥, drawn as a solid line, that includes points on the boundary itself.
- Overlap region
- The area where the shaded regions of every inequality in a system intersect, representing all shared solutions.
Watch it explained
- How it worksHow do you test a point against both inequalities of a system and demand that both hold?
- Worked exampleFind whether (4, 3) satisfies both y greater than x - 2 and 2x + y at most 12, one at a time.
- Worked exampleFind which of (1, 6), (5, 2), and (3, 9) lie in the overlap of y at least 2x - 4 and x + y under 10.
- Common mistakeWhy isn't a point sitting exactly on a dashed boundary line part of the solution set?
Check yourself
End of topic.