Question

solve the inequality \\(\frac{2}{3}x - \frac{5}{6} \ge \frac{1}{2}\\). which number line represents the graph of the solution?

Explanation:

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<reasoning>

Solve the inequality for x

\[

$$\begin{aligned} \frac{2}{3}x - \frac{5}{6} &\ge \frac{1}{2} \\ \frac{2}{3}x &\ge \frac{1}{2} + \frac{5}{6} \\ \frac{2}{3}x &\ge \frac{3}{6} + \frac{5}{6} \\ \frac{2}{3}x &\ge \frac{8}{6} \\ \frac{2}{3}x &\ge \frac{4}{3} \end{aligned}$$

\]

Isolate the variable x

\[

$$\begin{aligned} x &\ge \frac{4}{3} \cdot \frac{3}{2} \\ x &\ge 2 \end{aligned}$$

\]

Identify the matching number line graph

\[

$$\begin{aligned} &\text{The inequality } x \ge 2 \text{ is represented by:} \\ &\text{- A solid circle at } 2 \\ &\text{- A shaded arrow pointing to the right} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) Solid circle at 12, shaded to the right</mcq-option>
<mcq-correct>(B) Solid circle at 2, shaded to the right</mcq-correct>
<mcq-option>(C) Solid circle at \(-\frac{1}{2}\), shaded to the right</mcq-option>
<mcq-option>(D) Solid circle at \(-\frac{2}{9}\), shaded to the right</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2,
"fixed": true
}
},
{
"type": "arrow",
"params": [[2, 0], [5, 0]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 4
}
},
{
"type": "point",
"params": [[2, 0]],
"properties": {
"name": "2",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"labelPosition": [0, -15]
}
},
{
"type": "point",
"params": [[-2, 0]],
"properties": {
"name": "-2",
"size": 2,
"strokeColor": "#583C87",
"fillColor": "#583C87",
"withLabel": true,
"labelPosition": [0, -15]
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "0",
"size": 2,
"strokeColor": "#583C87",
"fillColor": "#583C87",
"withLabel": true,
"labelPosition": [0, -15]
}
},
{
"type": "point",
"params": [[4, 0]],
"properties": {
"name": "4",
"size": 2,
"strokeColor": "#583C87",
"fillColor": "#583C87",
"withLabel": true,
"labelPosition": [0, -15]
}
}
],
"timestamps": [0.5]
}
</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Solving Inequalities"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Solve the inequality for x

\[

$$\begin{aligned} \frac{2}{3}x - \frac{5}{6} &\ge \frac{1}{2} \\ \frac{2}{3}x &\ge \frac{1}{2} + \frac{5}{6} \\ \frac{2}{3}x &\ge \frac{3}{6} + \frac{5}{6} \\ \frac{2}{3}x &\ge \frac{8}{6} \\ \frac{2}{3}x &\ge \frac{4}{3} \end{aligned}$$

\]

Isolate the variable x

\[

$$\begin{aligned} x &\ge \frac{4}{3} \cdot \frac{3}{2} \\ x &\ge 2 \end{aligned}$$

\]

Identify the matching number line graph

\[

$$\begin{aligned} &\text{The inequality } x \ge 2 \text{ is represented by:} \\ &\text{- A solid circle at } 2 \\ &\text{- A shaded arrow pointing to the right} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) Solid circle at 12, shaded to the right</mcq-option>
<mcq-correct>(B) Solid circle at 2, shaded to the right</mcq-correct>
<mcq-option>(C) Solid circle at \(-\frac{1}{2}\), shaded to the right</mcq-option>
<mcq-option>(D) Solid circle at \(-\frac{2}{9}\), shaded to the right</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2,
"fixed": true
}
},
{
"type": "arrow",
"params": [[2, 0], [5, 0]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 4
}
},
{
"type": "point",
"params": [[2, 0]],
"properties": {
"name": "2",
"size": 5,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true,
"labelPosition": [0, -15]
}
},
{
"type": "point",
"params": [[-2, 0]],
"properties": {
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"size": 2,
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"labelPosition": [0, -15]
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"properties": {
"name": "4",
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"withLabel": true,
"labelPosition": [0, -15]
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}
],
"timestamps": [0.5]
}
</plot>

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"Mathematics",
"Algebra",
"Solving Inequalities"
]
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