systems of inequalities: practice

\
\begin{aligned}
x - y &\le -3 \\
2x + y &\ge 1
\end{aligned}
\

which solution is valid within the context of the situation?

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

Question

systems of inequalities: practice

\
\begin{aligned}
x - y &\le -3 \\
2x + y &\ge 1
\end{aligned}
\

which solution is valid within the context of the situation?

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

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" ] } ## Test the first point Using the *Systems of Linear Inequalities* and *Graphical Solutions* knowledge points $$ \begin{aligned} &\text{Point: } (-1.5, 4)\ &-1.5 - 4 = -5.5 \le -3 \quad (\text{True})\ &2(-1.5) + 4 = -3 + 4 = 1 \ge 1 \quad (\text{True}) \end{aligned} $$ ## Test the second point Using the *Systems of Linear Inequalities* and *Graphical Solutions* knowledge points $$ \begin{aligned} &\text{Point: } (-2, 1)\ &-2 - 1 = -3 \le -3 \quad (\text{True})\ &2(-2) + 1 = -4 + 1 = -3 \ge 1 \quad (\text{False}) \end{aligned} $$ ## Test the third point Using the *Systems of Linear Inequalities* and *Graphical Solutions* knowledge points $$ \begin{aligned} &\text{Point: } (1, 4.5)\ &1 - 4.5 = -3.5 \le -3 \quad (\text{True})\ &2(1) + 4.5 = 6.5 \ge 1 \quad (\text{True}) \end{aligned} $$ ## Test the fourth point Using the *Systems of Linear Inequalities* and *Graphical Solutions* knowledge points $$ \begin{aligned} &\text{Point: } (-1, 5)\ &-1 - 5 = -6 \le -3 \quad (\text{True})\ &2(-1) + 5 = 3 \ge 1 \quad (\text{True}) \end{aligned} $$ ## Identify the correct option from the visual graph Using the *Graphical Solutions* and *Feasible Region Interpretation* knowledge points $$ \begin{aligned} &\text{Feasible region is shaded in the upper-left quadrant.}\ &(-1.5, 4) \text{ lies clearly within the shaded region.}\ &(1, 4.5) \text{ lies outside the shaded region (to the right of } x - y = -3\text{).}\ &(-1, 5) \text{ lies on the boundary line } x - y = -3\text{ but is not listed as the primary solution.} \end{aligned} $$

(A) (-1.5, 4)

(B) (-2, 1)

(C) (1, 4.5)

(D) (-1, 5)

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Identify the correct option from the visual graph

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