Assessment item-review infographic

Linear Inequalities in Two Variables

Mathematics · Algebra I · English

Items10

PurposeSTAAR format practice and review

Draft for content-expert review and revision.

Item plan

Item	Standard	Evidence of learning	Stimulus	Format	Correct response
1	A.3(D)	Graph an inequality in slope-intercept form	None	Multiple choice	A
2	A.3(D)	Graph an inequality in standard form	None	Multiple choice	D
3	A.3(D)	Identify correct boundary and shading properties	None	Multiselect	A, C
4	A.3(D)	Describe boundary line type and region	None	Inline choice	solid, above
5	A.3(D)	Match inequalities to their graph properties	None	Match table grid	Row 1: dashed/below; Row 2: solid/above; Row 3: dashed/above
6	A.3(D)	Graph an inequality with a vertical boundary line	None	Multiple choice	B
7	A.3(D)	Select a coordinate point inside the solution region	Coordinate plane with dashed boundary and 4 points	Hot spot	Point D
8	A.3(D)	Construct the graph of a linear inequality	Empty coordinate plane	Graphing	Boundary: \(2x + y = -3\), shaded right/above
9	A.3(D)	Select the correct graph for a contextual situation	None	Multiple choice	C
10	A.3(D)	Identify intercepts and shading attributes	None	Drag and drop	(5, 0), (0, -4), dashed, includes

Item 1

Multiple choice

Which graph best represents the solution set of \(y > -2x + 4\)?

Select one correct answer.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student recognizes that the strict inequality (\(>\)) produces a dashed boundary line and that \(y >\) requires shading the region above the line.

Correct response

Response rationales
Response	Status	Reasoning and evidence
A	Correct	The line \(y = -2x + 4\) is graphed with a dashed line because the inequality is strict (\(>\)), and the region above the line is shaded because \(y\) is greater than the expression.
B	Distractor	The student correctly shaded above the line but used a solid boundary line, confusing \(>\) with \(\ge\).
C	Distractor	The student correctly used a dashed boundary line but shaded the region below the line, incorrectly interpreting \(>\).
D	Distractor	The student correctly used a dashed line and shaded above, but graphed an incorrect \(y\)-intercept of \(-4\) instead of \(4\).

Item 2

Multiple choice

Which graph represents the solution set of the inequality \(3x - 4y \le 12\)?

Select one correct answer.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student correctly solves for \(y\) (or uses test points) to identify the boundary line and the correct shading direction when the coefficient of \(y\) is negative.

Correct response

Response rationales
Response	Status	Reasoning and evidence
A	Distractor	The student graphed the correct boundary line but shaded the wrong region, likely because they did not flip the inequality symbol when dividing by \(-4\).
B	Distractor	The student graphed the correct intercepts but used a dashed line instead of a solid line, and shaded the incorrect region.
C	Distractor	The student incorrectly evaluated the \(y\)-intercept as \(3\) instead of \(-3\), plotting the line \(3x + 4y = 12\).
D	Correct	Converting to slope-intercept form gives \(y \ge \frac{3}{4}x - 3\). The boundary is a solid line passing through \((0, -3)\) and \((4, 0)\), and the solution set is the region above the line.

Item 3

Multiselect

Select TWO statements that correctly describe the graph of the solution set of the inequality \(x + 2y < 6\).

Select two correct answers.

The boundary line is dashed.
The boundary line is solid.
The shaded region includes the origin \((0, 0)\).
The shaded region includes the point \((4, 2)\).
The boundary line has a positive slope.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student evaluates attributes of the inequality's graph, including boundary line style and the validity of points in the shaded region.

Correct response

First and third statements (The boundary line is dashed; The shaded region includes the origin (0,0)).

Response rationales
Response	Status	Reasoning and evidence
The boundary line is dashed.	Correct	The inequality uses a strict less-than sign (\(<\)), meaning points on the line are not included in the solution set.
The boundary line is solid.	Distractor	The student confuses a strict inequality (\(<\)) with an inclusive inequality (\(\le\)).
The shaded region includes the origin \((0, 0)\).	Correct	Testing the origin gives \(0 + 2(0) < 6\), which simplifies to \(0 < 6\). Since this is true, the region containing \((0, 0)\) is shaded.
The shaded region includes the point \((4, 2)\).	Distractor	Testing the point \((4, 2)\) gives \(4 + 2(2) = 8\). Since \(8\) is not less than \(6\), this point is not in the shaded region.
The boundary line has a positive slope.	Distractor	Rewriting the equation \(x + 2y = 6\) in slope-intercept form gives \(y = -\frac{1}{2}x + 3\). The slope is negative, not positive.

Item 4

Inline choice

Choose the correct words to complete the statement about the inequality \(y - 3 \ge -2(x + 1)\).

Select the correct answer from each drop-down menu.

To graph the solution set of the inequality, the boundary line should be drawn as a line, and the shading should be placed the boundary line.

Menu 1: solid | dashed

Menu 2: above | below

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student determines the boundary style from the inequality symbol and determines the shading region from the \(y\)-isolation.

Correct response

Menu 1: solid; Menu 2: above

Response rationales
Response	Status	Reasoning and evidence
solid	Correct	The symbol is \(\ge\), indicating the boundary line points are solutions, so a solid line is required.
dashed	Distractor	The student associates the wrong line type with the \(\ge\) symbol.
above	Correct	The inequality states \(y\) is greater than or equal to the expression, which means values larger than the boundary line (above it) represent the solution set.
below	Distractor	The student incorrectly interprets the "greater than" relation or makes an error when simplifying the right side that flips their perception of the shading.

Item 5

Match table grid

Match each linear inequality to the correct description of its graph.

Select one correct answer in each row.

	Dashed line, shaded below	Dashed line, shaded above	Solid line, shaded below	Solid line, shaded above
\(y < \frac{1}{2}x - 3\)
\(x - 2y \le 6\)
\(x + 2y > -6\)

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student translates inequalities in standard form into their slope-intercept equivalent to identify shading direction and boundary type.

Correct response

Row 1: Dashed line, shaded below. Row 2: Solid line, shaded above. Row 3: Dashed line, shaded above.

Response rationales
Row	Correct classification	Reasoning and errors
\(y < \frac{1}{2}x - 3\)	Dashed line, shaded below	The symbol \(<\) represents a dashed line, and because \(y\) is isolated and less than the expression, the region below is shaded. Other options are incorrect applications of inequality symbols.
\(x - 2y \le 6\)	Solid line, shaded above	When solving for \(y\), dividing by \(-2\) reverses the inequality to \(y \ge \frac{1}{2}x - 3\). The \(\ge\) symbol requires a solid line and shading above. Students failing to flip the symbol will incorrectly choose "shaded below".
\(x + 2y > -6\)	Dashed line, shaded above	Solving for \(y\) gives \(y > -\frac{1}{2}x - 3\). The division is by positive \(2\), so the symbol \(>\) remains, requiring a dashed line and shading above. Errors result from erroneously flipping the inequality or misreading the strict inequality.

Item 6

Multiple choice

Which graph represents the solution set of \(x \ge -3\)?

Select one correct answer.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student graphs a one-variable linear inequality in the coordinate plane by recognizing it as a vertical boundary line.

Correct response

Response rationales
Response	Status	Reasoning and evidence
A	Distractor	The student graphed the boundary line correctly but shaded to the left, which represents \(x \le -3\).
B	Correct	The boundary line is vertical at \(x = -3\). Because the symbol is \(\ge\), the line is solid and the shaded region includes all \(x\)-values greater than \(-3\) (to the right).
C	Distractor	The student confused the \(x\) variable with the \(y\) variable, graphing the solution set for \(y \ge -3\).
D	Distractor	The student graphed a dashed boundary line instead of a solid line, confusing \(\ge\) with \(>\).

Item 7

Hot spot

The dashed boundary line for the inequality \(3x - 2y > 2\) is shown on the coordinate plane. Select the point that represents a solution to the inequality.

Select one location on the image.

Selectable hot spots: Point A, Point B, Point C, Point D.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student evaluates coordinate pairs against an inequality to determine which region contains the solution set relative to the boundary line.

Correct response

Point D (2, -2)

Response rationales
Response	Status	Reasoning and evidence
Point A (-2, 2)	Distractor	Testing \((-2, 2)\) yields \(3(-2) - 2(2) = -10\). Since \(-10\) is not greater than \(2\), it is not a solution. This region is incorrectly shaded if the student forgot to flip the inequality when dividing by \(-2\).
Point B (0, -1)	Distractor	Testing \((0, -1)\) yields \(3(0) - 2(-1) = 2\). Since \(2\) is not strictly greater than \(2\), points on the boundary line are not included.
Point C (4, 5)	Distractor	Testing \((4, 5)\) yields \(3(4) - 2(5) = 2\). This point lies exactly on the dashed boundary line and is not a solution.
Point D (2, -2)	Correct	Testing \((2, -2)\) yields \(3(2) - 2(-2) = 10\). Since \(10 > 2\), the point lies in the shaded region and is a valid solution.

Item 8

Graphing

Graph the solution set of the linear inequality \(y \ge -2x - 3\).

Plot the boundary line and shade the correct region.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student constructs the linear boundary with the correct slope and \(y\)-intercept, applies the correct line style, and identifies the correct solution half-plane.

Correct response

Solid boundary line through \((0, -3)\) and \((-1, -1)\); Shaded region above the line.

Completed correct graph

Response rationales
Component	Status	Reasoning and evidence
Boundary line	Required	The line must have a \(y\)-intercept of \(-3\) and a slope of \(-2\). The student plots points such as \((0, -3)\) and \((-1, -1)\).
Line style	Required	The inequality symbol is \(\ge\), so the boundary must be drawn as a solid line. A dashed line indicates the student incorrectly interpreted the inequality as strict.
Shaded region	Required	Because the inequality states \(y\) is greater than or equal to \(-2x - 3\), the region above the line (to the right) must be shaded. Shading below indicates an incorrect interpretation of the inequality symbol.

Item 9

Multiple choice

A baker has at most 12 cups of flour to make batches of cookies and muffins. Each batch of cookies requires 2 cups of flour, and each batch of muffins requires 3 cups of flour. Let \(x\) represent the number of batches of cookies and \(y\) represent the number of batches of muffins.

Which graph represents the possible combinations of batches of cookies and muffins the baker can make?

Select one correct answer.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student translates a real-world scenario into a linear inequality (\(2x + 3y \le 12\)) and identifies its corresponding graph.

Correct response

Response rationales
Response	Status	Reasoning and evidence
A	Distractor	The student incorrectly assigned the coefficients, graphing \(3x + 2y \le 12\) instead of \(2x + 3y \le 12\).
B	Distractor	The student used a dashed line, representing the strict inequality \(2x + 3y < 12\), instead of the inclusive "at most" condition (\(\le\)).
C	Correct	The situation is modeled by \(2x + 3y \le 12\). The \(x\)-intercept is \(6\), the \(y\)-intercept is \(4\), the line is solid (baker can use exactly 12 cups), and the shading is below the line representing values less than or equal to 12.
D	Distractor	The student graphed the correct boundary line but shaded above the line, which models \(2x + 3y \ge 12\) ("at least" 12 cups of flour).

Item 10

Drag and drop

Complete the description of the graph of the inequality \(4x - 5y < 20\).

Move the correct answer to each box.

\((5, 0)\) \((-5, 0)\) \((0, -4)\) \((0, 4)\) dashed solid includes does not include

The graph of \(4x - 5y < 20\) has an \(x\)-intercept of and a \(y\)-intercept of . The boundary line is and the shaded region the origin.

Writer review

Review required

Standard

A.3(D) · Graph the solution set of linear inequalities in two variables on the coordinate plane.

Evidence of learning

The student extracts necessary characteristics directly from the standard-form inequality to define its graph, including intercepts, line type, and test-point validity.

Correct response

Blank 1: \((5, 0)\); Blank 2: \((0, -4)\); Blank 3: dashed; Blank 4: includes

Response rationales
Blank	Correct answer	Reasoning and errors
1	\((5, 0)\)	Substituting \(y = 0\) yields \(4x = 20\), so the \(x\)-intercept is \(5\). Using \(-5\) reflects a sign error.
2	\((0, -4)\)	Substituting \(x = 0\) yields \(-5y = 20\), so the \(y\)-intercept is \(-4\). Using \(4\) reflects a sign error forgetting the negative coefficient.
3	dashed	The strict inequality symbol (\(<\)) means the boundary line values are not part of the solution, so the line is drawn dashed.
4	includes	Testing the origin \((0, 0)\) gives \(4(0) - 5(0) < 20\), which simplifies to \(0 < 20\). Since this is a true statement, the origin is included in the shaded region.