Question

graph the solution to the inequality on the number line.

\\|w - 7| \le 4\\

Explanation:

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<reasoning>

Set up the compound inequality

To solve the absolute value inequality, we rewrite it.
The inequality \(|w - 7| \le 4\) means the distance is at most \(4\).
This translates to:
\[-4 \le w - 7 \le 4\]

Solve for the variable

We isolate \(w\) by adding \(7\) to all parts.
\[-4 + 7 \le w \le 4 + 7\]
\[3 \le w \le 11\]

Determine the graph representation

The solution set is the closed interval \([3, 11]\).
On a number line, this is represented by:

• Solid circles at \(w = 3\) and \(w = 11\) to show inclusion.
• A shaded line segment connecting these two endpoints.

</reasoning>

<answer>
The solution to the inequality is \(3 \le w \le 11\).

To graph this on the number line:

• Place a solid circle at \(3\).
• Place a solid circle at \(11\).
• Draw a line segment connecting \(3\) and \(11\).

</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-12, 0], [12, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "segment",
"params": [[3, 0], [11, 0]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 6,
"fixed": true
}
},
{
"type": "point",
"params": [[3, 0]],
"properties": {
"name": "3",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"fixed": true
}
},
{
"type": "point",
"params": [[11, 0]],
"properties": {
"name": "11",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"fixed": true
}
}
],
"timestamps": [0.1]
}
</plot>

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"question_type": "Free Response",
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Answer:

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</pre_analysis>

<reasoning>

Set up the compound inequality

To solve the absolute value inequality, we rewrite it.
The inequality \(|w - 7| \le 4\) means the distance is at most \(4\).
This translates to:
\[-4 \le w - 7 \le 4\]

Solve for the variable

We isolate \(w\) by adding \(7\) to all parts.
\[-4 + 7 \le w \le 4 + 7\]
\[3 \le w \le 11\]

Determine the graph representation

The solution set is the closed interval \([3, 11]\).
On a number line, this is represented by:

• Solid circles at \(w = 3\) and \(w = 11\) to show inclusion.
• A shaded line segment connecting these two endpoints.

</reasoning>

<answer>
The solution to the inequality is \(3 \le w \le 11\).

To graph this on the number line:

• Place a solid circle at \(3\).
• Place a solid circle at \(11\).
• Draw a line segment connecting \(3\) and \(11\).

</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-12, 0], [12, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "segment",
"params": [[3, 0], [11, 0]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 6,
"fixed": true
}
},
{
"type": "point",
"params": [[3, 0]],
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"name": "3",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"fixed": true
}
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"type": "point",
"params": [[11, 0]],
"properties": {
"name": "11",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"fixed": true
}
}
],
"timestamps": [0.1]
}
</plot>

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"Mathematics",
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