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Sagot :
To solve the given system of inequalities and determine the graph representing the solution set, we'll proceed step-by-step through boundary line calculations and testing regions.
We start with the given system of inequalities:
1. [tex]\( 3y \geq x - 9 \)[/tex]
2. [tex]\( 3x + y > -3 \)[/tex]
### Step 1: Convert Inequalities to Equations
First, let's convert these inequalities into boundary lines.
#### For Inequality 1: [tex]\( 3y \geq x - 9 \)[/tex]
Rewriting it as an equation:
[tex]\[ 3y = x - 9 \rightarrow y = \frac{1}{3}x - 3 \][/tex]
#### For Inequality 2: [tex]\( 3x + y > -3 \)[/tex]
Rewriting it as an equation:
[tex]\[ 3x + y = -3 \rightarrow y = -3x - 3 \][/tex]
### Step 2: Identify Boundary Points
To graph these lines, we need a couple of points on each line to plot them accurately.
#### For the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{3}(0) - 3 = -3 \][/tex]
So, one point is [tex]\((0, -3)\)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{1}{3}(3) - 3 = -2 \][/tex]
So, another point is [tex]\((3, -2)\)[/tex].
#### For the line [tex]\( y = -3x - 3 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) - 3 = -3 \][/tex]
So, one point is [tex]\((0, -3)\)[/tex].
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -3(-1) - 3 = 0 \][/tex]
So, another point is [tex]\((-1, 0)\)[/tex].
### Step 3: Determine Regions to Shade
- For the inequality [tex]\( y \geq \frac{1}{3}x - 3 \)[/tex], the region above the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex] should be shaded.
- For the inequality [tex]\( y > -3x - 3 \)[/tex], the region above the line [tex]\( y = -3x - 3 \)[/tex] should be shaded.
### Step 4: Graphing and Testing a Point
To ensure these regions are correct, we can test a point that is not on either boundary line. Let's test the point [tex]\((0, 0)\)[/tex]:
- For [tex]\( 3y \geq x - 9 \)[/tex]:
[tex]\[ 3(0) \geq 0 - 9 \rightarrow 0 \geq -9 \][/tex] (True)
- For [tex]\( 3x + y > -3 \)[/tex]:
[tex]\[ 3(0) + 0 > -3 \rightarrow 0 > -3 \][/tex] (True)
Since [tex]\((0, 0)\)[/tex] satisfies both inequalities, it is within the solution region.
### Step 5: Plot Points and Boundaries
- Line 1: Passes through [tex]\((0, -3)\)[/tex] and [tex]\((3, -2)\)[/tex]
- Line 2: Passes through [tex]\((0, -3)\)[/tex] and [tex]\((-1, 0)\)[/tex]
### Summary:
Thus, the boundaries of the solution set are formed by [tex]\((0, -3)\)[/tex] and [tex]\((3, -2)\)[/tex] for the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex], and by [tex]\((0, -3)\)[/tex] and [tex]\((-1, 0)\)[/tex] for the line [tex]\( y = -3x - 3 \)[/tex].
The solution set is the region where the shaded areas of both inequalities overlap, considering the boundary lines and the regions above them:
- Boundary Line 1: [tex]\((0, -3)\)[/tex] to [tex]\((3, -2)\)[/tex]
- Boundary Line 2: [tex]\((0, -3)\)[/tex] to [tex]\((-1, 0)\)[/tex]
By plotting these points and the related lines on a graph, and shading the correct regions, we will get the overall solution set for the system of inequalities.
We start with the given system of inequalities:
1. [tex]\( 3y \geq x - 9 \)[/tex]
2. [tex]\( 3x + y > -3 \)[/tex]
### Step 1: Convert Inequalities to Equations
First, let's convert these inequalities into boundary lines.
#### For Inequality 1: [tex]\( 3y \geq x - 9 \)[/tex]
Rewriting it as an equation:
[tex]\[ 3y = x - 9 \rightarrow y = \frac{1}{3}x - 3 \][/tex]
#### For Inequality 2: [tex]\( 3x + y > -3 \)[/tex]
Rewriting it as an equation:
[tex]\[ 3x + y = -3 \rightarrow y = -3x - 3 \][/tex]
### Step 2: Identify Boundary Points
To graph these lines, we need a couple of points on each line to plot them accurately.
#### For the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{3}(0) - 3 = -3 \][/tex]
So, one point is [tex]\((0, -3)\)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{1}{3}(3) - 3 = -2 \][/tex]
So, another point is [tex]\((3, -2)\)[/tex].
#### For the line [tex]\( y = -3x - 3 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) - 3 = -3 \][/tex]
So, one point is [tex]\((0, -3)\)[/tex].
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -3(-1) - 3 = 0 \][/tex]
So, another point is [tex]\((-1, 0)\)[/tex].
### Step 3: Determine Regions to Shade
- For the inequality [tex]\( y \geq \frac{1}{3}x - 3 \)[/tex], the region above the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex] should be shaded.
- For the inequality [tex]\( y > -3x - 3 \)[/tex], the region above the line [tex]\( y = -3x - 3 \)[/tex] should be shaded.
### Step 4: Graphing and Testing a Point
To ensure these regions are correct, we can test a point that is not on either boundary line. Let's test the point [tex]\((0, 0)\)[/tex]:
- For [tex]\( 3y \geq x - 9 \)[/tex]:
[tex]\[ 3(0) \geq 0 - 9 \rightarrow 0 \geq -9 \][/tex] (True)
- For [tex]\( 3x + y > -3 \)[/tex]:
[tex]\[ 3(0) + 0 > -3 \rightarrow 0 > -3 \][/tex] (True)
Since [tex]\((0, 0)\)[/tex] satisfies both inequalities, it is within the solution region.
### Step 5: Plot Points and Boundaries
- Line 1: Passes through [tex]\((0, -3)\)[/tex] and [tex]\((3, -2)\)[/tex]
- Line 2: Passes through [tex]\((0, -3)\)[/tex] and [tex]\((-1, 0)\)[/tex]
### Summary:
Thus, the boundaries of the solution set are formed by [tex]\((0, -3)\)[/tex] and [tex]\((3, -2)\)[/tex] for the line [tex]\( y = \frac{1}{3}x - 3 \)[/tex], and by [tex]\((0, -3)\)[/tex] and [tex]\((-1, 0)\)[/tex] for the line [tex]\( y = -3x - 3 \)[/tex].
The solution set is the region where the shaded areas of both inequalities overlap, considering the boundary lines and the regions above them:
- Boundary Line 1: [tex]\((0, -3)\)[/tex] to [tex]\((3, -2)\)[/tex]
- Boundary Line 2: [tex]\((0, -3)\)[/tex] to [tex]\((-1, 0)\)[/tex]
By plotting these points and the related lines on a graph, and shading the correct regions, we will get the overall solution set for the system of inequalities.
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