Question

1. the graph of \\(y = -\frac{1}{4}x + 4\\) is shown on the grid.

choose the ordered pairs that are in the solution set of \\(y > -\frac{1}{4}x + 4\\).

\\((0, 0)\\) \\((-7, 9)\\) \\((0, 8)\\) \\((4, 3)\\) \\((9, 1)\\) \\((5, 5)\\)

Explanation:

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<reasoning>

Define the inequality and candidate points

\[
y > -\frac{1}{4}x + 4
\]
Candidate points: \((0,0)\), \((-7,9)\), \((0,8)\), \((4,3)\), \((9,1)\), \((5,5)\)

Evaluate each ordered pair

\[

$$\begin{aligned} (0,0): \quad & 0 > -\frac{1}{4}(0) + 4 \implies 0 > 4 \quad (\text{False}) \\ (-7,9): \quad & 9 > -\frac{1}{4}(-7) + 4 \implies 9 > 1.75 + 4 \implies 9 > 5.75 \quad (\text{True}) \\ (0,8): \quad & 8 > -\frac{1}{4}(0) + 4 \implies 8 > 4 \quad (\text{True}) \\ (4,3): \quad & 3 > -\frac{1}{4}(4) + 4 \implies 3 > -1 + 4 \implies 3 > 3 \quad (\text{False}) \\ (9,1): \quad & 1 > -\frac{1}{4}(9) + 4 \implies 1 > -2.25 + 4 \implies 1 > 1.75 \quad (\text{False}) \\ (5,5): \quad & 5 > -\frac{1}{4}(5) + 4 \implies 5 > -1.25 + 4 \implies 5 > 2.75 \quad (\text{True}) \end{aligned}$$

\]

Identify the valid solution set

The ordered pairs that satisfy the inequality are:
\[
(-7,9), \quad (0,8), \quad (5,5)
\]
</reasoning>

<answer>
\((-7,9)\), \((0,8)\), and \((5,5)\)
</answer>

<plot>
{
"elements": [
{
"type": "inequality",
"params": [
{
"js": "-0.25*x + 4",
"latex": "y > -\frac{1}{4}x + 4"
},
[0, 6]
],
"properties": {
"inverse": true,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
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"dash": 2
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},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "(0,0)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[-7, 9]],
"properties": {
"name": "(-7,9)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 8]],
"properties": {
"name": "(0,8)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 3]],
"properties": {
"name": "(4,3)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[9, 1]],
"properties": {
"name": "(9,1)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 5]],
"properties": {
"name": "(5,5)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

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Answer:

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<reasoning>

Define the inequality and candidate points

\[
y > -\frac{1}{4}x + 4
\]
Candidate points: \((0,0)\), \((-7,9)\), \((0,8)\), \((4,3)\), \((9,1)\), \((5,5)\)

Evaluate each ordered pair

\[

$$\begin{aligned} (0,0): \quad & 0 > -\frac{1}{4}(0) + 4 \implies 0 > 4 \quad (\text{False}) \\ (-7,9): \quad & 9 > -\frac{1}{4}(-7) + 4 \implies 9 > 1.75 + 4 \implies 9 > 5.75 \quad (\text{True}) \\ (0,8): \quad & 8 > -\frac{1}{4}(0) + 4 \implies 8 > 4 \quad (\text{True}) \\ (4,3): \quad & 3 > -\frac{1}{4}(4) + 4 \implies 3 > -1 + 4 \implies 3 > 3 \quad (\text{False}) \\ (9,1): \quad & 1 > -\frac{1}{4}(9) + 4 \implies 1 > -2.25 + 4 \implies 1 > 1.75 \quad (\text{False}) \\ (5,5): \quad & 5 > -\frac{1}{4}(5) + 4 \implies 5 > -1.25 + 4 \implies 5 > 2.75 \quad (\text{True}) \end{aligned}$$

\]

Identify the valid solution set

The ordered pairs that satisfy the inequality are:
\[
(-7,9), \quad (0,8), \quad (5,5)
\]
</reasoning>

<answer>
\((-7,9)\), \((0,8)\), and \((5,5)\)
</answer>

<plot>
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"elements": [
{
"type": "inequality",
"params": [
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"js": "-0.25*x + 4",
"latex": "y > -\frac{1}{4}x + 4"
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"params": [[0, 0]],
"properties": {
"name": "(0,0)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[-7, 9]],
"properties": {
"name": "(-7,9)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 8]],
"properties": {
"name": "(0,8)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 3]],
"properties": {
"name": "(4,3)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[9, 1]],
"properties": {
"name": "(9,1)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 5]],
"properties": {
"name": "(5,5)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

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