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Sagot :
Certainly! Let's go through the steps to graph the given linear inequality [tex]\(5x - y > -3\)[/tex] on the coordinate plane.
### Step-by-Step Solution to Graph [tex]\(5x - y > -3\)[/tex]:
1. Rewrite the inequality: We want to express the inequality in a way that helps us understand the boundary. The inequality [tex]\(5x - y > -3\)[/tex] can be rewritten by isolating [tex]\(y\)[/tex]:
[tex]\[ 5x - y > -3 \][/tex]
Adding [tex]\(y\)[/tex] to both sides and adding 3 to both sides, we get:
[tex]\[ 5x + 3 > y \][/tex]
This can be rewritten as:
[tex]\[ y < 5x + 3 \][/tex]
2. Graph the boundary line: The inequality [tex]\(y < 5x + 3\)[/tex] has a boundary line described by the equation [tex]\(y = 5x + 3\)[/tex].
- Since the inequality does not include equality (the [tex]\(<\)[/tex] symbol), we will graph the line [tex]\(y = 5x + 3\)[/tex] as a dashed line to represent that points on the line are not included in the solution.
3. Identify the region to shade: The inequality [tex]\(y < 5x + 3\)[/tex] tells us that the solution region is below the line [tex]\(y = 5x + 3\)[/tex].
4. Plot the boundary line:
- Find two points to plot the line:
1. When [tex]\(x = 0\)[/tex]:
[tex]\[ y = 5(0) + 3 = 3 \quad \text{(Point: } (0, 3) \text{)} \][/tex]
2. When [tex]\(x = 1\)[/tex]:
[tex]\[ y = 5(1) + 3 = 8 \quad \text{(Point: } (1, 8) \text{)} \][/tex]
- Plot these points on the coordinate plane and draw a dashed line through them.
5. Shade the appropriate region:
- Since the inequality is [tex]\(y < 5x + 3\)[/tex], shade the region below the dashed line.
### Summary
- Boundary Line: [tex]\(y = 5x + 3\)[/tex] (plot with a dashed line since [tex]\( < \)[/tex] does not include equality)
- Shading Region: Below the dashed line where [tex]\(y < 5x + 3\)[/tex]
### Visual Representation:
```
Coordinate Plane:
^
| / (1, 8)
| (dashed) /_____
|
| / /
| / / (Shaded Region)
| / /
| /_____(0, 3)
|
-----------------------------> x
```
In this visual, the dashed line represents the boundary [tex]\(y = 5x + 3\)[/tex], and the shading below the line shows the solution region where [tex]\(5x - y > -3\)[/tex], or equivalently [tex]\(y < 5x + 3\)[/tex].
### Step-by-Step Solution to Graph [tex]\(5x - y > -3\)[/tex]:
1. Rewrite the inequality: We want to express the inequality in a way that helps us understand the boundary. The inequality [tex]\(5x - y > -3\)[/tex] can be rewritten by isolating [tex]\(y\)[/tex]:
[tex]\[ 5x - y > -3 \][/tex]
Adding [tex]\(y\)[/tex] to both sides and adding 3 to both sides, we get:
[tex]\[ 5x + 3 > y \][/tex]
This can be rewritten as:
[tex]\[ y < 5x + 3 \][/tex]
2. Graph the boundary line: The inequality [tex]\(y < 5x + 3\)[/tex] has a boundary line described by the equation [tex]\(y = 5x + 3\)[/tex].
- Since the inequality does not include equality (the [tex]\(<\)[/tex] symbol), we will graph the line [tex]\(y = 5x + 3\)[/tex] as a dashed line to represent that points on the line are not included in the solution.
3. Identify the region to shade: The inequality [tex]\(y < 5x + 3\)[/tex] tells us that the solution region is below the line [tex]\(y = 5x + 3\)[/tex].
4. Plot the boundary line:
- Find two points to plot the line:
1. When [tex]\(x = 0\)[/tex]:
[tex]\[ y = 5(0) + 3 = 3 \quad \text{(Point: } (0, 3) \text{)} \][/tex]
2. When [tex]\(x = 1\)[/tex]:
[tex]\[ y = 5(1) + 3 = 8 \quad \text{(Point: } (1, 8) \text{)} \][/tex]
- Plot these points on the coordinate plane and draw a dashed line through them.
5. Shade the appropriate region:
- Since the inequality is [tex]\(y < 5x + 3\)[/tex], shade the region below the dashed line.
### Summary
- Boundary Line: [tex]\(y = 5x + 3\)[/tex] (plot with a dashed line since [tex]\( < \)[/tex] does not include equality)
- Shading Region: Below the dashed line where [tex]\(y < 5x + 3\)[/tex]
### Visual Representation:
```
Coordinate Plane:
^
| / (1, 8)
| (dashed) /_____
|
| / /
| / / (Shaded Region)
| / /
| /_____(0, 3)
|
-----------------------------> x
```
In this visual, the dashed line represents the boundary [tex]\(y = 5x + 3\)[/tex], and the shading below the line shows the solution region where [tex]\(5x - y > -3\)[/tex], or equivalently [tex]\(y < 5x + 3\)[/tex].
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