select the correct graph of the inequality:

(y \ge \frac{1}{3}x + 2)

a)
b)
c)
d)

a
b

Question

select the correct graph of the inequality:

(y \ge \frac{1}{3}x + 2)

a)
b)
c)
d)

a
b

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Linear Inequalities", "Graphing Inequalities", "Slope-Intercept Form" ], "new_concepts": [], "current_concepts": [ "Linear Inequalities", "Graphing Inequalities", "Slope-Intercept Form" ] } ## Identify the boundary line properties Using the *Slope-Intercept Form* knowledge point $$ \begin{aligned} &y = mx + b \implies y = \frac{1}{3}x + 2\ &\text{Slope } (m) = \frac{1}{3}\ &\text{y-intercept } (b) = 2 \implies (0, 2) \end{aligned} $$ ## Determine the boundary line type Using the *Linear Inequalities* knowledge point $$ \begin{aligned} &\text{Inequality symbol: } \ge\ &\text{Boundary line: solid (includes equality)} \end{aligned} $$ ## Determine the shaded region Using the *Graphing Inequalities* knowledge point $$ \begin{aligned} &\text{Test point: } (0,0)\ &0 \ge \frac{1}{3}(0) + 2 \implies 0 \ge 2 \quad (\text{False})\ &\text{Shade the region not containing } (0,0) \implies \text{above the line} \end{aligned} $$ ## Match with the correct option Using the *Graphing Inequalities* knowledge point $$ \begin{aligned} &\text{Graph A has a solid line with y-intercept } 2\text{, slope } \frac{1}{3}\text{, and shading above the line.}\ &\text{Therefore, Graph A is the correct representation.} \end{aligned} $$

(A) Graph A

(B) Graph B

(C) Graph C

(D) Graph D

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QUESTION IMAGE

Question

Explanation:

Identify the boundary line properties

Determine the boundary line type

Determine the shaded region

Match with the correct option

Answer: