graph the inequality.

\x + 3y

Question

graph the inequality.

\x + 3y < 3\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Graph Intercepts", "Linear Functions" ], "new_concepts": [ "Linear Inequalities", "Half-Plane Shading" ], "current_concepts": [ "Graph Intercepts", "Linear Functions", "Linear Inequalities", "Half-Plane Shading" ] } ## Find the boundary line intercepts $$ \begin{aligned} &x + 3y = 3 \ &\text{If } y = 0 \implies x = 3 \implies (3, 0) \ &\text{If } x = 0 \implies 3y = 3 \implies y = 1 \implies (0, 1) \end{aligned} $$ ## Determine the boundary line style and shaded region $$ \begin{aligned} &\text{Inequality: } x + 3y < 3 \ &\text{Strict inequality } (<) \implies \text{Dashed boundary line} \ &\text{Test point } (0,0): \ &0 + 3(0) = 0 < 3 \quad (\text{True}) \implies \text{Shade the half-plane containing } (0,0) \end{aligned} $$ To graph the inequality $x + 3y < 3$: 1. **Boundary Line**: Draw the dashed line $x + 3y = 3$ passing through the intercepts $(3, 0)$ and $(0, 1)$. 2. **Shading**: Shade the region below and to the left of the line, which contains the origin $(0, 0)$. { "elements": [ { "type": "inequality", "params": [ { "js": "(3 - x) / 3", "latex": "y < \frac{3 - x}{3}" }, [0, 0] ], "properties": { "inverse": false, "strict": true, "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2 } }, { "type": "point", "params": [[3, 0]], "properties": { "name": "(3,0)", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[0, 1]], "properties": { "name": "(0,1)", "size": 3, "color": "#583C87", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Linear Inequalities" ] }

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Question

Explanation:

Find the boundary line intercepts

Determine the boundary line style and shaded region

Answer: