Question

linear inequalities: mastery test

Explanation:

<pre_analysis>
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"new_concepts": [
"Linear Inequalities",
"Graphical Solutions"
],
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</pre_analysis>

<reasoning>

Identify the boundary line

The boundary line is dashed and passes through the y-intercept \((0, -1)\) and the x-intercept \((-3, 0)\).
Using the intercept form or slope-intercept form:
\[
\text{Slope } m = \frac{-1 - 0}{0 - (-3)} = -\frac{1}{3}
\]
\[
\text{Equation of the line: } y = -\frac{1}{3}x - 1
\]

Determine the inequality sign

The region above the dashed line is shaded.
Since the line is dashed, the inequality is strict (\(>\)).
\[
y > -\frac{1}{3}x - 1
\]

Convert to standard form

Multiply the entire inequality by 3 to clear the fraction:
\[
3y > -x - 3
\]
\[
x + 3y > -3
\]
</reasoning>

<answer>
\[x + 3y > -3\]
</answer>

<plot>
{
"elements": [
{
"type": "inequality",
"params": [
{
"js": "-1/3*x - 1",
"latex": "y > -\frac{1}{3}x - 1"
},
[0, 2]
],
"properties": {
"inverse": true,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"dash": 2
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},
{
"type": "point",
"params": [[0, -1]],
"properties": {
"name": "(0, -1)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
}
]
}
</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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"step_cot",
"explain_then_answer",
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],
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"new_concepts": [
"Linear Inequalities",
"Graphical Solutions"
],
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"Linear Inequalities",
"Graphical Solutions"
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</pre_analysis>

<reasoning>

Identify the boundary line

The boundary line is dashed and passes through the y-intercept \((0, -1)\) and the x-intercept \((-3, 0)\).
Using the intercept form or slope-intercept form:
\[
\text{Slope } m = \frac{-1 - 0}{0 - (-3)} = -\frac{1}{3}
\]
\[
\text{Equation of the line: } y = -\frac{1}{3}x - 1
\]

Determine the inequality sign

The region above the dashed line is shaded.
Since the line is dashed, the inequality is strict (\(>\)).
\[
y > -\frac{1}{3}x - 1
\]

Convert to standard form

Multiply the entire inequality by 3 to clear the fraction:
\[
3y > -x - 3
\]
\[
x + 3y > -3
\]
</reasoning>

<answer>
\[x + 3y > -3\]
</answer>

<plot>
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"elements": [
{
"type": "inequality",
"params": [
{
"js": "-1/3*x - 1",
"latex": "y > -\frac{1}{3}x - 1"
},
[0, 2]
],
"properties": {
"inverse": true,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.3)",
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"name": "(0, -1)",
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"color": "#583C87",
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{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
}
]
}
</plot>

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"question_type": "Free Response",
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"Mathematics",
"Algebra",
"Linear Inequalities"
]
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</post_analysis>