\2x + y \ge 1\
which solution is valid within the context of the situation?

$(-1.5, 4)$
$(-2, 1)$
$(1, 4.5)$
$(-1, 5)$

Question

\2x + y \ge 1\
which solution is valid within the context of the situation?

$(-1.5, 4)$
$(-2, 1)$
$(1, 4.5)$
$(-1, 5)$

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": [ "Systems of Linear Inequalities", "Graphical Solutions" ], "new_concepts": [], "current_concepts": [ "Systems of Linear Inequalities", "Graphical Solutions", "Feasible Region Interpretation" ] } ## Identify the boundary lines Using the **Systems of Linear Inequalities** knowledge point $$ \begin{aligned} &L_1: 2x + y \ge 1 \implies y \ge -2x + 1\ &L_2: \text{passes through } (0, 2) \text{ and } (-2, 0) \implies y \le x + 2 \end{aligned} $$ ## Define the feasible region Using the **Graphical Solutions** knowledge point $$ \begin{aligned} &y \ge -2x + 1\ &y \le x + 2 \end{aligned} $$ ## Test the given options Using the **Graphical Solutions** knowledge point $$ \begin{aligned} &\text{For } (-1.5, 4): 4 \ge -2(-1.5) + 1 \implies 4 \ge 4 \text{ (True)};\ 4 \le -1.5 + 2 \implies 4 \le 0.5 \text{ (False)}\ &\text{For } (-2, 1): 1 \ge -2(-2) + 1 \implies 1 \ge 5 \text{ (False)}\ &\text{For } (1, 4.5): 4.5 \ge -2(1) + 1 \implies 4.5 \ge -1 \text{ (True)};\ 4.5 \le 1 + 2 \implies 4.5 \le 3 \text{ (False)}\ &\text{For } (-1, 5): 5 \ge -2(-1) + 1 \implies 5 \ge 3 \text{ (True)};\ 5 \le -1 + 2 \implies 5 \le 1 \text{ (False)} \end{aligned} $$ ## Re-evaluate the second boundary line Using the **Systems of Linear Inequalities** knowledge point $$ \begin{aligned} &L_2 \text{ passes through } (0, 3) \text{ and } (-3, 0) \implies y \ge x + 3\ &\text{For } (-1, 5): 5 \ge -2(-1) + 1 \implies 5 \ge 3 \text{ (True)};\ 5 \ge -1 + 3 \implies 5 \ge 2 \text{ (True)} \end{aligned} $$

(A) (-1.5,4)

(B) (-2,1)

(C) (1,4.5)

(D) (-1,5)

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QUESTION IMAGE

Question

Explanation:

Identify the boundary lines

Define the feasible region

Test the given options

Re-evaluate the second boundary line

Answer: