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Sagot :
Let's solve this system of inequalities step by step and graph the solution on the coordinate plane.
### Step 1: Graph the Boundary Lines
First, we'll graph the boundary lines for each inequality.
#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]
The boundary line for this inequality is:
[tex]\[ y = 2x + 4 \][/tex]
To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 4 = 4 \][/tex]
So, the point [tex]\((0, 4)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2(1) + 4 = 6 \][/tex]
So, the point [tex]\((1, 6)\)[/tex].
Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex], and draw a dashed line through them (since the inequality is [tex]\( > \)[/tex], not [tex]\( \geq \)[/tex]).
#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]
The boundary line for this inequality is:
[tex]\[ x + y = 6 \][/tex]
To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 6 \][/tex]
So, the point [tex]\((0, 6)\)[/tex].
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x = 6 \][/tex]
So, the point [tex]\((6, 0)\)[/tex].
Plot the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex], and draw a solid line through them (since the inequality is [tex]\( \leq \)[/tex]).
### Step 2: Shade the Solution Areas
Now, we need to determine which regions to shade for each inequality.
#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]
Since it’s [tex]\( y > 2x + 4 \)[/tex], we shade the region above the line [tex]\( y = 2x + 4 \)[/tex].
#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]
Since it’s [tex]\( x + y \leq 6 \)[/tex], we shade the region below or on the line [tex]\( x + y = 6 \)[/tex].
### Step 3: Find the Intersection of the Shaded Areas
The solution to the system of inequalities will be the region where the shaded areas overlap. Here's what you should see on the graph:
1. The line [tex]\( y = 2x + 4 \)[/tex] (dashed) going through the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] with shading above this line.
2. The line [tex]\( x + y = 6 \)[/tex] (solid) going through the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex] with shading below or on this line.
The overlapping shaded region represents the solution set to the system of inequalities.
### Step 1: Graph the Boundary Lines
First, we'll graph the boundary lines for each inequality.
#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]
The boundary line for this inequality is:
[tex]\[ y = 2x + 4 \][/tex]
To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 4 = 4 \][/tex]
So, the point [tex]\((0, 4)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2(1) + 4 = 6 \][/tex]
So, the point [tex]\((1, 6)\)[/tex].
Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex], and draw a dashed line through them (since the inequality is [tex]\( > \)[/tex], not [tex]\( \geq \)[/tex]).
#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]
The boundary line for this inequality is:
[tex]\[ x + y = 6 \][/tex]
To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 6 \][/tex]
So, the point [tex]\((0, 6)\)[/tex].
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x = 6 \][/tex]
So, the point [tex]\((6, 0)\)[/tex].
Plot the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex], and draw a solid line through them (since the inequality is [tex]\( \leq \)[/tex]).
### Step 2: Shade the Solution Areas
Now, we need to determine which regions to shade for each inequality.
#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]
Since it’s [tex]\( y > 2x + 4 \)[/tex], we shade the region above the line [tex]\( y = 2x + 4 \)[/tex].
#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]
Since it’s [tex]\( x + y \leq 6 \)[/tex], we shade the region below or on the line [tex]\( x + y = 6 \)[/tex].
### Step 3: Find the Intersection of the Shaded Areas
The solution to the system of inequalities will be the region where the shaded areas overlap. Here's what you should see on the graph:
1. The line [tex]\( y = 2x + 4 \)[/tex] (dashed) going through the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] with shading above this line.
2. The line [tex]\( x + y = 6 \)[/tex] (solid) going through the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex] with shading below or on this line.
The overlapping shaded region represents the solution set to the system of inequalities.
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