Graph systems of linear inequalities in two variables, identify the overlapping solution region, and determine whether a given point is a solution (A.EI.5).

What this topic is asking

A.EI.5 asks you to graph a system of linear inequalities in two variables, find the overlapping solution region, and decide whether a point is a solution. On the Virginia Algebra I SOL these are Equations and Inequalities items: pick the correct graph, identify the solution region, or test a point. They appear as multiple choice, coordinate-plane (shading) items, and hot-spot point tests.

Graphing one linear inequality

Graphing $y < 2x + 1$ (or any linear inequality) has three parts:

1. Graph the boundary line $y = 2x + 1$ as if it were an equation.
2. Choose the line style. Solid for $\le$ or $\ge$ (the line's points are solutions); dashed for $<$ or $>$ (they are not).
3. Shade the correct side. Pick a test point not on the line, usually $(0, 0)$, and substitute. If it makes the inequality true, shade that side; if false, shade the other side.

For $y < 2x + 1$, test $(0, 0)$: $0 < 1$ is true, so shade the side containing the origin (below the line).

A system is the overlap

A system of inequalities asks for the points that satisfy all of them simultaneously. Graph each inequality on the same plane, and the solution region is where the shadings overlap. A point in the overlap satisfies every inequality; a point in only one shaded region does not.

Testing whether a point is a solution

To check a candidate point, substitute it into every inequality. It is a solution only if all of them are true. If the point fails even one, it is not in the region. For boundaries, remember that a solid line includes its points (so equality counts) while a dashed line excludes them.

Solid versus dashed, and why it matters

The boundary style encodes whether the edge of the region belongs to the solution, and it follows directly from the inequality symbol. An inclusive symbol ($\le$, $\ge$) means the points where the two sides are equal are allowed, so the boundary is part of the solution and is drawn solid. A strict symbol ($<$, $>$) means equality is not allowed, so the boundary is excluded and drawn dashed. This is the same open-versus-closed-circle idea from one-variable inequalities, lifted to two dimensions: a closed circle becomes a solid line, an open circle a dashed line. Getting the style right matters on the SOL because a point sitting exactly on a boundary is a solution only when that boundary is solid, and coordinate-plane items often place a test point right on the edge to check whether you know the difference.

Modeling with a system

Constraint problems become systems of inequalities. "You can spend at most \20$, with apples at$\$2$ and bananas at \1$, buying at least$3$apples" gives$2a + b \le 20$and$a \ge 3$, plus the natural conditions$a \ge 0$,$b \ge 0$. The solution region (often called the feasible region) is every combination that meets all the constraints, and any point inside it is an allowed purchase.

How the SOL examines this topic

• Multiple choice. Identify the boundary style, the shaded region, or the solution of a system.
• Coordinate-plane items. Graph one or two inequalities and shade the overlap.
• Hot spot / point test. Decide whether a marked point lies in the solution region.

Try this

Q1. Should the boundary of $y \ge 3x - 2$ be solid or dashed? [1 point]

• Cue. Inclusive ($\ge$), so solid.

Q2. Is $(0, 0)$ a solution of $y > x + 1$? [1 point]

• Cue. $0 > 1$ is false, so no.