Question

click on a tool to begin drawing.

\\2x + y > 4\\

Explanation:

<pre_analysis>
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"Graphing Linear Inequalities"
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<reasoning>

Rearrange the inequality to slope-intercept form

\[

$$\begin{aligned} 2x + y &> 4 \\ y &> -2x + 4 \end{aligned}$$

\]

Identify boundary line properties

\[

$$\begin{aligned} &\text{Boundary Line Equation: } y = -2x + 4 \\ &\text{y-intercept: } (0, 4) \\ &\text{x-intercept: } (2, 0) \\ &\text{Line Style: Dashed (due to strict inequality } >) \end{aligned}$$

\]

Determine the shaded region

\[

$$\begin{aligned} &\text{Test Point: } (0,0) \\ &2(0) + 0 = 0 gtr 4 \\ &\text{Shade the region above/to the right of the line } y = -2x + 4 \end{aligned}$$

\]
</reasoning>

<answer>
To graph the inequality \(2x + y > 4\):

1. Plot the boundary line \(y = -2x + 4\) as a dashed line passing through the points \((0, 4)\) and \((2, 0)\).
2. Shade the region above and to the right of this boundary line (the region that does not contain the origin \((0,0)\)).

</answer>

<plot>
{
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{
"type": "inequality",
"params": [
{
"js": "-2*x + 4",
"latex": "y > -2x + 4"
},
[4, 4]
],
"properties": {
"inverse": true,
"strict": true,
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"fillColor": "rgba(140, 85, 242, 0.3)"
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{
"type": "point",
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</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Inequalities"
]
}
</post_analysis>

Answer:

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"Graphing Linear Inequalities"
],
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</pre_analysis>

<reasoning>

Rearrange the inequality to slope-intercept form

\[

$$\begin{aligned} 2x + y &> 4 \\ y &> -2x + 4 \end{aligned}$$

\]

Identify boundary line properties

\[

$$\begin{aligned} &\text{Boundary Line Equation: } y = -2x + 4 \\ &\text{y-intercept: } (0, 4) \\ &\text{x-intercept: } (2, 0) \\ &\text{Line Style: Dashed (due to strict inequality } >) \end{aligned}$$

\]

Determine the shaded region

\[

$$\begin{aligned} &\text{Test Point: } (0,0) \\ &2(0) + 0 = 0 gtr 4 \\ &\text{Shade the region above/to the right of the line } y = -2x + 4 \end{aligned}$$

\]
</reasoning>

<answer>
To graph the inequality \(2x + y > 4\):

1. Plot the boundary line \(y = -2x + 4\) as a dashed line passing through the points \((0, 4)\) and \((2, 0)\).
2. Shade the region above and to the right of this boundary line (the region that does not contain the origin \((0,0)\)).

</answer>

<plot>
{
"elements": [
{
"type": "inequality",
"params": [
{
"js": "-2*x + 4",
"latex": "y > -2x + 4"
},
[4, 4]
],
"properties": {
"inverse": true,
"strict": true,
"strokeColor": "#8C55F2",
"strokeWidth": 2,
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"color": "#583C87",
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}
}
],
"timestamps": [0.5]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Inequalities"
]
}
</post_analysis>