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Sagot :
Certainly! Let's solve and graph the inequality [tex]\(2x + 3y > 12\)[/tex] step by step.
### Step 1: Understand the Inequality
The inequality [tex]\(2x + 3y > 12\)[/tex] means that we need to find all the points [tex]\((x, y)\)[/tex] which, when substituted into the expression on the left side, result in a value greater than 12.
### Step 2: Find the Boundary Line
The boundary line for the inequality [tex]\(2x + 3y > 12\)[/tex] is the equation:
[tex]\[ 2x + 3y = 12 \][/tex]
We'll use this boundary line to divide the plane into two regions: one where [tex]\(2x + 3y \leq 12\)[/tex] and one where [tex]\(2x + 3y > 12\)[/tex].
### Step 3: Plot the Boundary Line
To plot the boundary line, find the intercepts:
1. x-intercept: Set [tex]\(y = 0\)[/tex] and solve for [tex]\(x\)[/tex]:
[tex]\[ 2x + 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
So, the x-intercept is [tex]\((6, 0)\)[/tex].
2. y-intercept: Set [tex]\(x = 0\)[/tex] and solve for [tex]\(y\)[/tex]:
[tex]\[ 2(0) + 3y = 12 \implies 3y = 12 \implies y = 4 \][/tex]
So, the y-intercept is [tex]\((0, 4)\)[/tex].
Now, plot these intercepts on the Cartesian plane and draw the line passing through the points [tex]\((6, 0)\)[/tex] and [tex]\((0, 4)\)[/tex].
### Step 4: Determine the Shading Region
The inequality [tex]\(2x + 3y > 12\)[/tex] indicates that we need to shade the region above this boundary line (since we want values greater than 12).
- To verify which side to shade, you can check a test point:
- Using the point [tex]\((0, 0)\)[/tex]:
[tex]\[ 2(0) + 3(0) = 0 \quad \text{(which is not greater than 12)} \][/tex]
- Since [tex]\((0, 0)\)[/tex] is not in the solution region, we shade the side of the boundary line that does not include this point.
### Step 5: Graphically Shade the Area
1. Draw the line [tex]\(2x + 3y = 12\)[/tex] solidly to indicate that the line itself is not included in the inequality solution.
2. Shade the region above the line to represent all the points where [tex]\(2x + 3y > 12\)[/tex].
### Final Graph Interpretation
The graph should look like this:
1. Boundary line: A straight line passing through points [tex]\((6, 0)\)[/tex] and [tex]\((0, 4)\)[/tex].
2. Shaded region: All the area above this line.
By plotting it on the Cartesian plane and shading the correct region, you will visually represent the solution to the inequality [tex]\(2x + 3y > 12\)[/tex].
### Step 1: Understand the Inequality
The inequality [tex]\(2x + 3y > 12\)[/tex] means that we need to find all the points [tex]\((x, y)\)[/tex] which, when substituted into the expression on the left side, result in a value greater than 12.
### Step 2: Find the Boundary Line
The boundary line for the inequality [tex]\(2x + 3y > 12\)[/tex] is the equation:
[tex]\[ 2x + 3y = 12 \][/tex]
We'll use this boundary line to divide the plane into two regions: one where [tex]\(2x + 3y \leq 12\)[/tex] and one where [tex]\(2x + 3y > 12\)[/tex].
### Step 3: Plot the Boundary Line
To plot the boundary line, find the intercepts:
1. x-intercept: Set [tex]\(y = 0\)[/tex] and solve for [tex]\(x\)[/tex]:
[tex]\[ 2x + 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
So, the x-intercept is [tex]\((6, 0)\)[/tex].
2. y-intercept: Set [tex]\(x = 0\)[/tex] and solve for [tex]\(y\)[/tex]:
[tex]\[ 2(0) + 3y = 12 \implies 3y = 12 \implies y = 4 \][/tex]
So, the y-intercept is [tex]\((0, 4)\)[/tex].
Now, plot these intercepts on the Cartesian plane and draw the line passing through the points [tex]\((6, 0)\)[/tex] and [tex]\((0, 4)\)[/tex].
### Step 4: Determine the Shading Region
The inequality [tex]\(2x + 3y > 12\)[/tex] indicates that we need to shade the region above this boundary line (since we want values greater than 12).
- To verify which side to shade, you can check a test point:
- Using the point [tex]\((0, 0)\)[/tex]:
[tex]\[ 2(0) + 3(0) = 0 \quad \text{(which is not greater than 12)} \][/tex]
- Since [tex]\((0, 0)\)[/tex] is not in the solution region, we shade the side of the boundary line that does not include this point.
### Step 5: Graphically Shade the Area
1. Draw the line [tex]\(2x + 3y = 12\)[/tex] solidly to indicate that the line itself is not included in the inequality solution.
2. Shade the region above the line to represent all the points where [tex]\(2x + 3y > 12\)[/tex].
### Final Graph Interpretation
The graph should look like this:
1. Boundary line: A straight line passing through points [tex]\((6, 0)\)[/tex] and [tex]\((0, 4)\)[/tex].
2. Shaded region: All the area above this line.
By plotting it on the Cartesian plane and shading the correct region, you will visually represent the solution to the inequality [tex]\(2x + 3y > 12\)[/tex].
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