Question

7 the graph of \\(7x - 4y = 8\\) is shown on the grid.

which points are in the solution set of \\(7x - 4y < 8\\)?

select two correct answers.

Explanation:

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<reasoning>

Identify the given inequality and points

Using the Linear Inequalities and Ordered Pairs knowledge points
\[

$$\begin{aligned} &\text{Inequality: } 7x - 4y < 8\\ &P_1 = (-4, 1)\\ &P_2 = (-1, 3)\\ &P_3 = (3, -4)\\ &P_4 = (5, 3) \end{aligned}$$

\]

Test the first point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (-4, 1):\\ &7(-4) - 4(1) = -28 - 4 = -32\\ &-32 < 8 \quad (\text{True}) \end{aligned}$$

\]

Test the second point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (-1, 3):\\ &7(-1) - 4(3) = -7 - 12 = -19\\ &-19 < 8 \quad (\text{True}) \end{aligned}$$

\]

Test the third point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (3, -4):\\ &7(3) - 4(-4) = 21 + 16 = 37\\ &37 ot< 8 \quad (\text{False}) \end{aligned}$$

\]

Test the fourth point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (5, 3):\\ &7(5) - 4(3) = 35 - 12 = 23\\ &23 ot< 8 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \((-4, 1)\)</mcq-correct>
<mcq-correct>(B) \((-1, 3)\)</mcq-correct>
<mcq-option>(C) \((3, -4)\)</mcq-option>
<mcq-option>(D) \((5, 3)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[0, -2], [4, 5]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 3,
"name": "7x - 4y = 8",
"withLabel": true,
"labelPosition": [3, 2]
}
},
{
"type": "point",
"params": [[-4, 1]],
"properties": {
"name": "(-4, 1)",
"size": 4,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-1, 3]],
"properties": {
"name": "(-1, 3)",
"size": 4,
"strokeColor": "#8C55F2",
"fillColor": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[3, -4]],
"properties": {
"name": "(3, -4)",
"size": 4,
"strokeColor": "#F2557F",
"fillColor": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 3]],
"properties": {
"name": "(5, 3)",
"size": 4,
"strokeColor": "#F2557F",
"fillColor": "#F2557F",
"withLabel": true
}
},
{
"type": "inequality",
"params": [{"js": "1.75*x - 2", "latex": "y > 1.75x - 2"}, [0, 2]],
"properties": {
"inverse": true,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.15)"
}
}
]
}
</plot>

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Answer:

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<reasoning>

Identify the given inequality and points

Using the Linear Inequalities and Ordered Pairs knowledge points
\[

$$\begin{aligned} &\text{Inequality: } 7x - 4y < 8\\ &P_1 = (-4, 1)\\ &P_2 = (-1, 3)\\ &P_3 = (3, -4)\\ &P_4 = (5, 3) \end{aligned}$$

\]

Test the first point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (-4, 1):\\ &7(-4) - 4(1) = -28 - 4 = -32\\ &-32 < 8 \quad (\text{True}) \end{aligned}$$

\]

Test the second point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (-1, 3):\\ &7(-1) - 4(3) = -7 - 12 = -19\\ &-19 < 8 \quad (\text{True}) \end{aligned}$$

\]

Test the third point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (3, -4):\\ &7(3) - 4(-4) = 21 + 16 = 37\\ &37 ot< 8 \quad (\text{False}) \end{aligned}$$

\]

Test the fourth point

Using the Ordered Pairs knowledge point
\[

$$\begin{aligned} &\text{Substitute } (5, 3):\\ &7(5) - 4(3) = 35 - 12 = 23\\ &23 ot< 8 \quad (\text{False}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \((-4, 1)\)</mcq-correct>
<mcq-correct>(B) \((-1, 3)\)</mcq-correct>
<mcq-option>(C) \((3, -4)\)</mcq-option>
<mcq-option>(D) \((5, 3)\)</mcq-option>
</answer>

<plot>
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"type": "line",
"params": [[0, -2], [4, 5]],
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"type": "point",
"params": [[-4, 1]],
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"name": "(-4, 1)",
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{
"type": "point",
"params": [[-1, 3]],
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"name": "(-1, 3)",
"size": 4,
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"withLabel": true
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},
{
"type": "point",
"params": [[3, -4]],
"properties": {
"name": "(3, -4)",
"size": 4,
"strokeColor": "#F2557F",
"fillColor": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 3]],
"properties": {
"name": "(5, 3)",
"size": 4,
"strokeColor": "#F2557F",
"fillColor": "#F2557F",
"withLabel": true
}
},
{
"type": "inequality",
"params": [{"js": "1.75*x - 2", "latex": "y > 1.75x - 2"}, [0, 2]],
"properties": {
"inverse": true,
"strict": true,
"fillColor": "rgba(140, 85, 242, 0.15)"
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]
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