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Sagot :
To solve the system of inequalities [tex]\( y + 2x > 3 \)[/tex] and [tex]\( y \geq 3.5x - 5 \)[/tex]:
1. Convert the first inequality to slope-intercept form:
The first inequality is originally given as [tex]\( y + 2x > 3 \)[/tex].
To convert this to slope-intercept form, solve for [tex]\( y \)[/tex]:
[tex]\[ y + 2x > 3 \implies y > -2x + 3 \][/tex]
The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
2. Determine the type of boundary line for the first inequality:
The inequality [tex]\( y > -2x + 3 \)[/tex] is a strict inequality (does not include the equal sign).
This means that the boundary line corresponding to this inequality is dashed (it represents points that are not part of the solution set).
3. Determine the type of boundary line for the second inequality:
The second inequality is given as [tex]\( y \geq 3.5x - 5 \)[/tex].
Since it includes the equal sign ([tex]\( \geq \)[/tex]), the boundary line for this inequality is solid (it represents points that are part of the solution set).
4. Determine the region to shade for both inequalities:
In the case of inequality [tex]\( y > -2x + 3 \)[/tex], we shade the region above the line [tex]\( y = -2x + 3 \)[/tex].
As for [tex]\( y \geq 3.5x - 5 \)[/tex], we also shade the region above the line [tex]\( y = 3.5x - 5 \)[/tex].
5. Check whether a specific point is in the solution set:
Let's check the point [tex]\( (0, 0) \)[/tex]:
- For the first inequality [tex]\( y + 2x > 3 \)[/tex]:
[tex]\[ 0 + 2 \cdot 0 > 3 \implies 0 > 3 \quad \text{(False)} \][/tex]
- For the second inequality [tex]\( y \geq 3.5x - 5 \)[/tex]:
[tex]\[ 0 \geq 3.5 \cdot 0 - 5 \implies 0 \geq -5 \quad \text{(True)} \][/tex]
Since the point [tex]\( (0, 0) \)[/tex] does not satisfy the first inequality, it is not in the solution set of the system of inequalities.
Finally, putting it all together:
- The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
- The first inequality, [tex]\( y + 2x > 3 \)[/tex], has a dashed boundary line.
- The second inequality, [tex]\( y \geq 3.5x - 5 \)[/tex], has a solid boundary line.
- Both inequalities have a solution set that is shaded above their boundary lines.
- There does not exist a point, such as [tex]\( (0, 0) \)[/tex], in the solution set of the system of inequalities.
Therefore, the solution for the given question is:
```
The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
The first inequality, [tex]\( y + 2x > 3 \)[/tex], has a dashed boundary line.
The second inequality, [tex]\( y \geq 3.5x - 5 \)[/tex], has a solid boundary line.
Both inequalities have a solution set that is shaded above their boundary lines.
None is a point in the solution set of the system of inequalities.
```
1. Convert the first inequality to slope-intercept form:
The first inequality is originally given as [tex]\( y + 2x > 3 \)[/tex].
To convert this to slope-intercept form, solve for [tex]\( y \)[/tex]:
[tex]\[ y + 2x > 3 \implies y > -2x + 3 \][/tex]
The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
2. Determine the type of boundary line for the first inequality:
The inequality [tex]\( y > -2x + 3 \)[/tex] is a strict inequality (does not include the equal sign).
This means that the boundary line corresponding to this inequality is dashed (it represents points that are not part of the solution set).
3. Determine the type of boundary line for the second inequality:
The second inequality is given as [tex]\( y \geq 3.5x - 5 \)[/tex].
Since it includes the equal sign ([tex]\( \geq \)[/tex]), the boundary line for this inequality is solid (it represents points that are part of the solution set).
4. Determine the region to shade for both inequalities:
In the case of inequality [tex]\( y > -2x + 3 \)[/tex], we shade the region above the line [tex]\( y = -2x + 3 \)[/tex].
As for [tex]\( y \geq 3.5x - 5 \)[/tex], we also shade the region above the line [tex]\( y = 3.5x - 5 \)[/tex].
5. Check whether a specific point is in the solution set:
Let's check the point [tex]\( (0, 0) \)[/tex]:
- For the first inequality [tex]\( y + 2x > 3 \)[/tex]:
[tex]\[ 0 + 2 \cdot 0 > 3 \implies 0 > 3 \quad \text{(False)} \][/tex]
- For the second inequality [tex]\( y \geq 3.5x - 5 \)[/tex]:
[tex]\[ 0 \geq 3.5 \cdot 0 - 5 \implies 0 \geq -5 \quad \text{(True)} \][/tex]
Since the point [tex]\( (0, 0) \)[/tex] does not satisfy the first inequality, it is not in the solution set of the system of inequalities.
Finally, putting it all together:
- The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
- The first inequality, [tex]\( y + 2x > 3 \)[/tex], has a dashed boundary line.
- The second inequality, [tex]\( y \geq 3.5x - 5 \)[/tex], has a solid boundary line.
- Both inequalities have a solution set that is shaded above their boundary lines.
- There does not exist a point, such as [tex]\( (0, 0) \)[/tex], in the solution set of the system of inequalities.
Therefore, the solution for the given question is:
```
The first inequality, [tex]\( y + 2x > 3 \)[/tex], is [tex]\( y > -2x + 3 \)[/tex] in slope-intercept form.
The first inequality, [tex]\( y + 2x > 3 \)[/tex], has a dashed boundary line.
The second inequality, [tex]\( y \geq 3.5x - 5 \)[/tex], has a solid boundary line.
Both inequalities have a solution set that is shaded above their boundary lines.
None is a point in the solution set of the system of inequalities.
```
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