Graph linear inequalities in two variables using a boundary line and shading, and find the solution region of a system of linear inequalities (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard moves inequalities into the coordinate plane. A linear inequality in two variables like $y > 2x - 3$ has a whole half-plane of solutions, not a single line, and the Georgia Milestones EOC asks you to graph it with the correct boundary line and shading, and then to find the overlap region for a system of inequalities. These show up as hot-spot and graphing items (shade the region, identify the boundary) and as multiple-choice items asking whether a point lies in the solution set. The two decisions to get right are the boundary line style and which side to shade.

Graphing a single inequality

The process has three steps.

1. Graph the boundary line by treating the inequality as an equation.
2. Choose the line style: solid for $\le$ or $\ge$ (boundary points are solutions), dashed for $<$ or $>$ (boundary points are not).
3. Shade the correct half-plane using a test point.

The test-point method

The reliable way to decide which side to shade is to test a point not on the boundary. The origin $(0, 0)$ is easiest when the line does not pass through it. Substitute it into the inequality:

• If the result is true, shade the side containing that point.
• If the result is false, shade the other side.

This works for any inequality and avoids guessing from the inequality direction, which can be misleading after rearranging.

Systems of inequalities

A system of linear inequalities is solved by the region that satisfies all of them at once. Graph each inequality on the same axes, and the solution is the overlap of the shaded regions.

To test whether a specific point is a solution, check it in each inequality; it qualifies only if all are satisfied.

How the Milestones examines this topic

• Hot spot / graphing. Identify the boundary line style (solid versus dashed) and shade the correct half-plane, or shade the overlap for a system.
• Multiple choice. Determine whether a point lies in the solution region of a system.
• Constructed response. Write and graph a system of inequalities modeling a constraint, then interpret the feasible region.

Why the line style encodes the boundary

The solid-versus-dashed choice is not decoration; it records whether the points exactly on the boundary count. For $y \ge 2x - 3$, a point on the line satisfies the inequality with equality, so those points are solutions and the line is drawn solid. For $y > 2x - 3$, a point on the line gives $y = 2x - 3$, which is not strictly greater, so boundary points are excluded and the line is dashed. This is the two-variable version of the open-versus-closed-circle rule on a number line, and it matters in a modeling context: if a constraint is "no more than" a limit, the limit itself is allowed (solid), but "strictly less than" excludes it (dashed). Reading whether the boundary is attainable is the same interpretive skill across one and two variables.

Inequalities as constraints

In modeling, each inequality is a constraint, and the overlap region is the set of feasible choices. "You have at most $40 to spend; sandwiches cost$5 and drinks cost $2" gives$5s + 2d \le 40$, with$s \ge 0$and$d \ge 0$because counts cannot be negative. The feasible region is the set of affordable, sensible combinations, and the corners of that region are where the best choices often sit. Recognizing that the non-negativity conditions$s \ge 0$and$d \ge 0$ are part of the system (they keep the region in the first quadrant) is a common omission the EOC checks.

Try this

Q1. For $y \le -x + 4$, state the boundary line style and a test for shading. [1 point]

• Cue. Solid line ($\le$); test $(0,0)$: $0 \le 4$ true, so shade the side with the origin (below).

Q2. Is $(1, 1)$ a solution of $\begin{cases} y < 2x \\ y \ge 0 \end{cases}$? [1 point]

• Cue. $1 < 2(1)$ true and $1 \ge 0$ true, so yes.