Graph a system of linear inequalities in two variables and identify the solution as the overlap of the half-planes, including testing whether a point lies in the solution region (Ohio A-REI.12).

What this topic is asking

Ohio standard A-REI.12 extends graphing one inequality to a system of two (or more) linear inequalities. Each inequality shades a half-plane; the solution of the system is the region where the half-planes overlap, the points that satisfy all the inequalities at once. This is the graphical heart of modeling with constraints and appears in the Expressions and Equations category.

Graphing the overlap

Graph one inequality, then the next, and find where the shadings agree.

Testing a point against the whole system

The single most reliable check is substitution into every inequality.

So testing is an AND, not an OR: all conditions must hold together. This mirrors the graph, the overlap is exactly the set of points inside all the shaded regions.

Bounded and unbounded regions

Two inequalities usually overlap in an unbounded region (a wedge open to infinity). Add more inequalities, or constraints like $x \geq 0$ and $y \geq 0$, and the overlap can close into a bounded polygon, the feasible region used in modeling. The corner points of that polygon, where boundary lines cross, matter when a model asks for a maximum or minimum.

How Ohio examines this topic

• Graphing. Shade two half-planes and identify the overlap on the coordinate grid.
• Equation response or multiple-select. Decide whether given points lie in the solution region by testing all inequalities.
• Multiple choice. Match a system to the graph of its overlapping region.

Why the solution is the intersection of the regions

Each inequality's solution set is every point that makes it true, an entire half-plane. A point solves the system only if it is true for all the inequalities simultaneously, which means it must lie in all of those half-planes at once. The set of points common to all of them is, by definition, their intersection, the overlap. That is why a point on the correct side of one boundary but the wrong side of another is excluded: membership in the solution set requires every condition, and the picture of "every condition" is the region where all the shadings coincide.

Why each edge keeps its own line style

When you read the boundary of the overlap, each edge inherits the solid-or-dashed style of the inequality that produced it. An edge from a $\geq$ inequality is solid (its boundary points are solutions), while an edge from a strict $>$ inequality is dashed (its boundary points are excluded). So a single feasible region can have some solid edges and some dashed edges. Keeping each edge's style correct matters on exact-match graphing items and when deciding whether a corner point itself counts as a solution.

Try this

Q1. Is $(2, 0)$ a solution of $y \leq x$ and $y \geq -1$? [2 points]

• Cue. $0 \leq 2$ true and $0 \geq -1$ true, so yes.

Q2. Two inequalities both use $\leq$. Are the boundary edges of the overlap solid or dashed? [1 point]

• Cue. Inclusive, so solid edges.