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Sagot :
Sure, let's solve the given system of inequalities graphically and state the coordinates of a point in the solution set. The steps to solve this problem are as follows:
### Step 1: Graph the Boundary Lines
First, we need to graph the boundary lines of the given inequalities:
#### Inequality 1: [tex]\( y \geq x - 8 \)[/tex]
The boundary line is [tex]\( y = x - 8 \)[/tex].
- To plot this, we need at least two points. Let's find where it intersects the x-axis and y-axis.
- Intersection with y-axis (x=0): [tex]\( y = 0 - 8 \Rightarrow y = -8 \)[/tex]
- Intersection with x-axis (y=0): [tex]\( 0 = x - 8 \Rightarrow x = 8 \)[/tex]
Points: [tex]\((0, -8)\)[/tex] and [tex]\((8, 0)\)[/tex]
#### Inequality 2: [tex]\( y < -1.5x - 3 \)[/tex]
The boundary line is [tex]\( y = -1.5x - 3 \)[/tex].
- Again, we need two points.
- Intersection with y-axis (x=0): [tex]\( y = -1.5(0) - 3 \Rightarrow y = -3 \)[/tex]
- Intersection with x-axis (y=0): [tex]\( 0 = -1.5x - 3 \Rightarrow 1.5x = -3 \Rightarrow x = -2 \)[/tex]
Points: [tex]\((0, -3)\)[/tex] and [tex]\((-2, 0)\)[/tex]
### Step 2: Plot the Boundary Lines and Shading
- Plot the line [tex]\( y = x - 8 \)[/tex] on the graph. Since the inequality is [tex]\( y \geq x - 8 \)[/tex], we will shade the area above this line.
- Plot the line [tex]\( y = -1.5x - 3 \)[/tex] on the graph. Since the inequality is [tex]\( y < -1.5x - 3 \)[/tex], we will shade the area below this line.
### Step 3: Identify the Solution Set
The solution set to the system of inequalities is the region where the shaded areas overlap.
### Step 4: Find the Coordinates of a Point within the Solution Set
We need to identify a specific point within the solution set. A good approach is to choose an x-value within the overlapping region and check for a corresponding y-value that satisfies both inequalities.
Let's choose [tex]\( x = -7 \)[/tex]:
1. For [tex]\( y \geq x - 8 \)[/tex]:
[tex]\[ y \geq -7 - 8 \Rightarrow y \geq -15 \][/tex]
2. For [tex]\( y < -1.5x - 3 \)[/tex]:
[tex]\[ y < -1.5(-7) - 3 \Rightarrow y < 10.5 - 3 \Rightarrow y < 7.5 \][/tex]
So, for [tex]\( x = -7 \)[/tex], [tex]\( y \)[/tex] should satisfy:
[tex]\[ -15 \leq y < 7.5 \][/tex]
One possible value for [tex]\( y \)[/tex] within this range is [tex]\( y = -10 \)[/tex].
Thus, the point [tex]\((-7, -10)\)[/tex] lies within the solution set of the given system of inequalities.
### Conclusion
The coordinates of a point in the solution set are [tex]\((-7, -10)\)[/tex].
### Step 1: Graph the Boundary Lines
First, we need to graph the boundary lines of the given inequalities:
#### Inequality 1: [tex]\( y \geq x - 8 \)[/tex]
The boundary line is [tex]\( y = x - 8 \)[/tex].
- To plot this, we need at least two points. Let's find where it intersects the x-axis and y-axis.
- Intersection with y-axis (x=0): [tex]\( y = 0 - 8 \Rightarrow y = -8 \)[/tex]
- Intersection with x-axis (y=0): [tex]\( 0 = x - 8 \Rightarrow x = 8 \)[/tex]
Points: [tex]\((0, -8)\)[/tex] and [tex]\((8, 0)\)[/tex]
#### Inequality 2: [tex]\( y < -1.5x - 3 \)[/tex]
The boundary line is [tex]\( y = -1.5x - 3 \)[/tex].
- Again, we need two points.
- Intersection with y-axis (x=0): [tex]\( y = -1.5(0) - 3 \Rightarrow y = -3 \)[/tex]
- Intersection with x-axis (y=0): [tex]\( 0 = -1.5x - 3 \Rightarrow 1.5x = -3 \Rightarrow x = -2 \)[/tex]
Points: [tex]\((0, -3)\)[/tex] and [tex]\((-2, 0)\)[/tex]
### Step 2: Plot the Boundary Lines and Shading
- Plot the line [tex]\( y = x - 8 \)[/tex] on the graph. Since the inequality is [tex]\( y \geq x - 8 \)[/tex], we will shade the area above this line.
- Plot the line [tex]\( y = -1.5x - 3 \)[/tex] on the graph. Since the inequality is [tex]\( y < -1.5x - 3 \)[/tex], we will shade the area below this line.
### Step 3: Identify the Solution Set
The solution set to the system of inequalities is the region where the shaded areas overlap.
### Step 4: Find the Coordinates of a Point within the Solution Set
We need to identify a specific point within the solution set. A good approach is to choose an x-value within the overlapping region and check for a corresponding y-value that satisfies both inequalities.
Let's choose [tex]\( x = -7 \)[/tex]:
1. For [tex]\( y \geq x - 8 \)[/tex]:
[tex]\[ y \geq -7 - 8 \Rightarrow y \geq -15 \][/tex]
2. For [tex]\( y < -1.5x - 3 \)[/tex]:
[tex]\[ y < -1.5(-7) - 3 \Rightarrow y < 10.5 - 3 \Rightarrow y < 7.5 \][/tex]
So, for [tex]\( x = -7 \)[/tex], [tex]\( y \)[/tex] should satisfy:
[tex]\[ -15 \leq y < 7.5 \][/tex]
One possible value for [tex]\( y \)[/tex] within this range is [tex]\( y = -10 \)[/tex].
Thus, the point [tex]\((-7, -10)\)[/tex] lies within the solution set of the given system of inequalities.
### Conclusion
The coordinates of a point in the solution set are [tex]\((-7, -10)\)[/tex].
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