Question

the following system of inequalities shows the relationship between two numbers, where the value of \\(x\\) is less than the value of \\(y\\) and both are integers.

\\
\

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

\\

which solution is valid within the context of the situation?

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

Explanation:

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<reasoning>

Identify the constraints

Using the Real-world Constraints knowledge point

• The problem states that the numbers must be integers.
• Therefore, any valid coordinate pair \((x, y)\) must consist of integer values.

Evaluate the options for integer constraints

Using the Real-world Constraints knowledge point

• \((-1.5, 4)\): Contains a non-integer \(x = -1.5\).
• \((-2, 1)\): Both coordinates are integers.
• \((1, 4.5)\): Contains a non-integer \(y = 4.5\).
• \((-1, 5)\): Both coordinates are integers.

Test the remaining integer options in the system

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &\text{System:} \quad \begin{cases} x - y \le -3 \\ 2x + y \ge 1 \end{cases} \end{aligned}$$

\]

• For \((-2, 1)\):

\[

$$\begin{aligned} &-2 - 1 = -3 \le -3 \quad (\text{True}) \\ &2(-2) + 1 = -3 \ge 1 \quad (\text{False}) \end{aligned}$$

\]

• For \((-1, 5)\):

\[

$$\begin{aligned} &-1 - 5 = -6 \le -3 \quad (\text{True}) \\ &2(-1) + 5 = 3 \ge 1 \quad (\text{True}) \end{aligned}$$

\]

Verify the valid solution graphically

Using the Graphical Solutions knowledge point

• The point \((-1, 5)\) lies within the shaded region of the graph.
• The point \((-2, 1)\) lies outside the shaded region.

</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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{
"elements": [
{
"type": "inequality",
"params": [{"js": "x + 3", "latex": "x - y \le -3"}, [-2, 4]],
"properties": {
"inverse": true,
"strict": false,
"strokeColor": "#583C87",
"fillColor": "rgba(140, 85, 242, 0.15)"
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{
"type": "inequality",
"params": [{"js": "1 - 2*x", "latex": "2x + y \ge 1"}, [1, 2]],
"properties": {
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"type": "point",
"params": [[-1, 5]],
"properties": {
"name": "(-1, 5)",
"size": 4,
"color": "#F2557F",
"withLabel": true
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"type": "point",
"params": [[-2, 1]],
"properties": {
"name": "(-2, 1)",
"size": 4,
"color": "#55DDF2",
"withLabel": true
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}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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Answer:

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<reasoning>

Identify the constraints

Using the Real-world Constraints knowledge point

• The problem states that the numbers must be integers.
• Therefore, any valid coordinate pair \((x, y)\) must consist of integer values.

Evaluate the options for integer constraints

Using the Real-world Constraints knowledge point

• \((-1.5, 4)\): Contains a non-integer \(x = -1.5\).
• \((-2, 1)\): Both coordinates are integers.
• \((1, 4.5)\): Contains a non-integer \(y = 4.5\).
• \((-1, 5)\): Both coordinates are integers.

Test the remaining integer options in the system

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &\text{System:} \quad \begin{cases} x - y \le -3 \\ 2x + y \ge 1 \end{cases} \end{aligned}$$

\]

• For \((-2, 1)\):

\[

$$\begin{aligned} &-2 - 1 = -3 \le -3 \quad (\text{True}) \\ &2(-2) + 1 = -3 \ge 1 \quad (\text{False}) \end{aligned}$$

\]

• For \((-1, 5)\):

\[

$$\begin{aligned} &-1 - 5 = -6 \le -3 \quad (\text{True}) \\ &2(-1) + 5 = 3 \ge 1 \quad (\text{True}) \end{aligned}$$

\]

Verify the valid solution graphically

Using the Graphical Solutions knowledge point

• The point \((-1, 5)\) lies within the shaded region of the graph.
• The point \((-2, 1)\) lies outside the shaded region.

</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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"size": 4,
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</plot>

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