Graph the solution set of linear inequalities in two variables on the coordinate plane, using a solid or dashed boundary and shading the correct half-plane (TEKS A.3D).

What this topic is asking

TEKS A.3D asks you to graph the solution set of a linear inequality in two variables. On the redesigned STAAR this appears as a traditional question (match the description or the graph) and as a hot-spot item (select the matching inequality or shade a region). Two decisions define the graph: the boundary line style (solid or dashed) and which half-plane to shade.

Step one: the boundary line and its style

Replace the inequality symbol with an equals sign and graph that line. The symbol decides the style:

• $\le$ or $\ge$: solid line, because points on the line satisfy the inequality.
• $<$ or $>$: dashed line, because points on the line are not solutions.

So $y \ge x + 2$ uses a solid line; $y < x + 2$ uses a dashed line. Choosing the wrong style is a frequent error on hot-spot items, where the solid-versus-dashed choice is part of the answer.

Step two: which half-plane to shade

Once the boundary is drawn, shade the region containing all the solution points. Two reliable methods:

• Solve for $y$. With $y$ isolated, shade above for $>$ or $\ge$ and below for $<$ or $\le$.
• Test a point. Pick a point not on the line, often $(0, 0)$, and substitute. If the inequality is true, shade that point's side; if false, shade the other side.

Why the test point works

A boundary line splits the plane into two half-planes, and a linear inequality is true on exactly one of them. So a single test point settles the whole region: whichever side that point lands on, the truth value of the inequality there is the truth value for the entire half-plane. Choosing the origin is convenient whenever the line does not pass through it.

How STAAR examines this topic

• Multiple choice. Describe the graph (dashed or solid, above or below), or match an inequality to a shaded graph.
• Hot spot. Select the inequality that matches a shown graph, or identify the boundary and region. Both the line style and the shading must be right.
• In systems. Two-variable inequalities reappear when shading the overlap region for a system of inequalities.

A clarifying idea is that the boundary is the set of equality cases, so a solid line literally includes those edge points as solutions while a dashed line marks them as the cutoff that is not reached.

Inequalities in context and vertical or horizontal boundaries

In a real-world constraint, the shaded region is the set of feasible combinations. If a worker earns $10 per lawn ($x$) and$15 per car wash ($y$) and needs at least $120, the constraint$10x + 15y \ge 120$describes every acceptable workload, and only the part in the first quadrant (non-negative$x$and$y$) makes sense. Restricting to sensible values is the interpretive step a context item rewards.

Boundaries are not always slanted. A horizontal boundary comes from an inequality like $y \le 4$ (shade everything on or below the horizontal line $y = 4$), and a vertical boundary comes from $x > 2$ (a dashed vertical line $x = 2$, shaded to the right). The same two decisions apply: the symbol sets solid versus dashed, and a test point or the direction of the inequality sets the shaded side. These simpler boundaries are easy points when you do not over-think them.

Try this

Q1. State the boundary style and shading for $y > -x + 1$. [1 point]

• Cue. Dashed line (strict), shaded above.

Q2. Is $(1, 1)$ a solution of $3x - y \le 5$? [1 point]

• Cue. $3(1) - 1 = 2 \le 5$ is true, so yes.