Question

systems of inequalities: practice

which system of inequalities has the following graph as its solution?

\\(\circ \

$$\begin{cases} x > 3 \\\\ y \\ge 2x - 5 \\end{cases}$$

\\)

\\(\circ \

$$\begin{cases} x > 3 \\\\ y \\le 2x - 5 \\end{cases}$$

\\)

\\(\circ \

$$\begin{cases} x < 3 \\\\ y \\ge 2x - 5 \\end{cases}$$

\\)

\\(\circ \

$$\begin{cases} x < 3 \\\\ y \\le 2x - 5 \\end{cases}$$

\\)

Explanation:

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<reasoning>

Identify the vertical boundary line

\[

$$\begin{aligned} &\text{Boundary line: } x = 3 \quad (\text{dashed line})\\ &\text{Shaded region is to the left of } x = 3\\ &\text{Inequality: } x < 3 \end{aligned}$$

\]

Identify the slanted boundary line

\[

$$\begin{aligned} &\text{Boundary line: } y = 2x - 5 \quad (\text{solid line})\\ &\text{Shaded region is above/left of the line}\\ &\text{Test point } (3, 3) \text{ in shaded region: } 3 \ge 2(3) - 5 \implies 3 \ge 1 \quad (\text{True})\\ &\text{Inequality: } y \ge 2x - 5 \end{aligned}$$

\]

Combine the inequalities

\[

$$\begin{cases} x < 3 \\ y \ge 2x - 5 \end{cases}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(

$$\begin{cases} x > 3 \\ y \ge 2x - 5 \end{cases}$$

\)</mcq-option>
<mcq-option>(B) \(

$$\begin{cases} x > 3 \\ y \le 2x - 5 \end{cases}$$

\)</mcq-option>
<mcq-correct>(C) \(

$$\begin{cases} x < 3 \\ y \ge 2x - 5 \end{cases}$$

\)</mcq-correct>
<mcq-option>(D) \(

$$\begin{cases} x < 3 \\ y \le 2x - 5 \end{cases}$$

\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the vertical boundary line

\[

$$\begin{aligned} &\text{Boundary line: } x = 3 \quad (\text{dashed line})\\ &\text{Shaded region is to the left of } x = 3\\ &\text{Inequality: } x < 3 \end{aligned}$$

\]

Identify the slanted boundary line

\[

$$\begin{aligned} &\text{Boundary line: } y = 2x - 5 \quad (\text{solid line})\\ &\text{Shaded region is above/left of the line}\\ &\text{Test point } (3, 3) \text{ in shaded region: } 3 \ge 2(3) - 5 \implies 3 \ge 1 \quad (\text{True})\\ &\text{Inequality: } y \ge 2x - 5 \end{aligned}$$

\]

Combine the inequalities

\[

$$\begin{cases} x < 3 \\ y \ge 2x - 5 \end{cases}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(

$$\begin{cases} x > 3 \\ y \ge 2x - 5 \end{cases}$$

\)</mcq-option>
<mcq-option>(B) \(

$$\begin{cases} x > 3 \\ y \le 2x - 5 \end{cases}$$

\)</mcq-option>
<mcq-correct>(C) \(

$$\begin{cases} x < 3 \\ y \ge 2x - 5 \end{cases}$$

\)</mcq-correct>
<mcq-option>(D) \(

$$\begin{cases} x < 3 \\ y \le 2x - 5 \end{cases}$$

\)</mcq-option>
</answer>

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