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)\\)

Explanation:

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<reasoning>

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
\[

$$\begin{aligned} &L_2 \text{ passes through } (0, 3) \text{ and } (-3, 0) \implies y \ge x + 3\\ &\text{For } (-1, 5): 5 \ge -2(-1) + 1 \implies 5 \ge 3 \text{ (True)};\ 5 \ge -1 + 3 \implies 5 \ge 2 \text{ (True)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) (-1.5,4)</mcq-option>
<mcq-option>(B) (-2,1)</mcq-option>
<mcq-option>(C) (1,4.5)</mcq-option>
<mcq-correct>(D) (-1,5)</mcq-correct>
</answer>

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"type": "inequality",
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Answer:

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<reasoning>

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
\[

$$\begin{aligned} &L_2 \text{ passes through } (0, 3) \text{ and } (-3, 0) \implies y \ge x + 3\\ &\text{For } (-1, 5): 5 \ge -2(-1) + 1 \implies 5 \ge 3 \text{ (True)};\ 5 \ge -1 + 3 \implies 5 \ge 2 \text{ (True)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) (-1.5,4)</mcq-option>
<mcq-option>(B) (-2,1)</mcq-option>
<mcq-option>(C) (1,4.5)</mcq-option>
<mcq-correct>(D) (-1,5)</mcq-correct>
</answer>

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