Question

find the graph of this inequality.

\\y < -x - 1\\

Explanation:

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<reasoning>

Identify the boundary line properties

Using the Slope-Intercept Form and y-intercept knowledge points
\[

$$\begin{aligned} &\text{Boundary line: } y = -x - 1\\ &\text{Slope } (m) = -1\\ &\text{y-intercept } (b) = -1 \implies (0, -1)\\ &\text{x-intercept: } 0 = -x - 1 \implies x = -1 \implies (-1, 0) \end{aligned}$$

\]

Determine the line style

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Inequality symbol: } <\\ &\text{Since it is strict (no "or equal to"), the boundary line must be dashed.} \end{aligned}$$

\]

Determine the shaded region

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Test point: } (0,0)\\ &0 < -0 - 1 \implies 0 < -1 \quad (\text{False})\\ &\text{Shade the region below/left of the line (not containing } (0,0)\text{).} \end{aligned}$$

\]

Match with the options

\[

$$\begin{aligned} &\text{Option 1: Solid line, shaded above/right. (Incorrect)}\\ &\text{Option 2: Dashed line, shaded below/left. (Correct)}\\ &\text{Option 3: Solid line, shaded below/left. (Incorrect)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) Solid line, shaded above and to the right</mcq-option>
<mcq-correct>(B) Dashed line, shaded below and to the left</mcq-correct>
<mcq-option>(C) Solid line, shaded below and to the left</mcq-option>
</answer>

<plot>
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"elements": [
{
"type": "inequality",
"params": [
{
"js": "-x - 1",
"latex": "y < -x - 1"
},
[-2, -2]
],
"properties": {
"inverse": false,
"strict": true,
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"fillColor": "rgba(140, 85, 242, 0.3)"
}
},
{
"type": "point",
"params": [[0, -1]],
"properties": {
"name": "(0, -1)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
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{
"type": "point",
"params": [[-1, 0]],
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"name": "(-1, 0)",
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"color": "#583C87",
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"timestamps": [0.5]
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</plot>

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Answer:

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<reasoning>

Identify the boundary line properties

Using the Slope-Intercept Form and y-intercept knowledge points
\[

$$\begin{aligned} &\text{Boundary line: } y = -x - 1\\ &\text{Slope } (m) = -1\\ &\text{y-intercept } (b) = -1 \implies (0, -1)\\ &\text{x-intercept: } 0 = -x - 1 \implies x = -1 \implies (-1, 0) \end{aligned}$$

\]

Determine the line style

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Inequality symbol: } <\\ &\text{Since it is strict (no "or equal to"), the boundary line must be dashed.} \end{aligned}$$

\]

Determine the shaded region

Using the Solving Inequalities knowledge point
\[

$$\begin{aligned} &\text{Test point: } (0,0)\\ &0 < -0 - 1 \implies 0 < -1 \quad (\text{False})\\ &\text{Shade the region below/left of the line (not containing } (0,0)\text{).} \end{aligned}$$

\]

Match with the options

\[

$$\begin{aligned} &\text{Option 1: Solid line, shaded above/right. (Incorrect)}\\ &\text{Option 2: Dashed line, shaded below/left. (Correct)}\\ &\text{Option 3: Solid line, shaded below/left. (Incorrect)} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) Solid line, shaded above and to the right</mcq-option>
<mcq-correct>(B) Dashed line, shaded below and to the left</mcq-correct>
<mcq-option>(C) Solid line, shaded below and to the left</mcq-option>
</answer>

<plot>
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"type": "inequality",
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"js": "-x - 1",
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},
[-2, -2]
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"inverse": false,
"strict": true,
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"fillColor": "rgba(140, 85, 242, 0.3)"
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{
"type": "point",
"params": [[0, -1]],
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"name": "(0, -1)",
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"timestamps": [0.5]
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</plot>

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