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To determine which system of inequalities has no solution, we need to analyze each system one by one. Let's consider the three given systems and evaluate whether there are values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy each set of inequalities simultaneously.
### System 1
[tex]\[ \begin{cases} x + 3y \geq 0.5 \\ x + 3y \leq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \geq 0.5 \)[/tex]
2. [tex]\( x + 3y \leq 2.5 \)[/tex]
Combining these inequalities, we get:
[tex]\[ 0.5 \leq x + 3y \leq 2.5 \][/tex]
This forms a bounded region where [tex]\( x + 3y \)[/tex] must lie between 0.5 and 2.5. It implies there are valid solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both inequalities simultaneously.
### System 2
[tex]\[ \begin{cases} x + 3y \geq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \geq 0.5 \)[/tex]
2. [tex]\( x + 3y \geq 2.5 \)[/tex]
Combining these inequalities, we see that the second inequality, [tex]\( x + 3y \geq 2.5 \)[/tex], is stricter. If [tex]\( x + 3y \geq 2.5 \)[/tex], it certainly satisfies [tex]\( x + 3y \geq 0.5 \)[/tex] as well. Thus, we only need [tex]\( x + 3y \geq 2.5 \)[/tex].
This means the system will have solutions as long as [tex]\( x + 3y \geq 2.5 \)[/tex]. There is a region where both inequalities hold true, implying valid solutions exist.
### System 3
[tex]\[ \begin{cases} x + 3y \leq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \leq 0.5 \)[/tex]
2. [tex]\( x + 3y \geq 2.5 \)[/tex]
Combining these inequalities, we see that they form a contradiction:
[tex]\[ x + 3y \leq 0.5 \][/tex]
[tex]\[ x + 3y \geq 2.5 \][/tex]
Here's why it's a contradiction:
- The first inequality requires [tex]\( x + 3y \)[/tex] to be 0.5 or less.
- The second inequality requires [tex]\( x + 3y \)[/tex] to be 2.5 or more.
Both conditions cannot be true simultaneously for any [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Therefore, this system has no solutions.
### Conclusion
Based on the analysis:
- System 1 has solutions.
- System 2 has solutions.
- System 3 has no solutions.
Therefore, the system of inequalities with no solution is:
[tex]\[ \begin{cases} x + 3y \leq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
### System 1
[tex]\[ \begin{cases} x + 3y \geq 0.5 \\ x + 3y \leq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \geq 0.5 \)[/tex]
2. [tex]\( x + 3y \leq 2.5 \)[/tex]
Combining these inequalities, we get:
[tex]\[ 0.5 \leq x + 3y \leq 2.5 \][/tex]
This forms a bounded region where [tex]\( x + 3y \)[/tex] must lie between 0.5 and 2.5. It implies there are valid solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both inequalities simultaneously.
### System 2
[tex]\[ \begin{cases} x + 3y \geq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \geq 0.5 \)[/tex]
2. [tex]\( x + 3y \geq 2.5 \)[/tex]
Combining these inequalities, we see that the second inequality, [tex]\( x + 3y \geq 2.5 \)[/tex], is stricter. If [tex]\( x + 3y \geq 2.5 \)[/tex], it certainly satisfies [tex]\( x + 3y \geq 0.5 \)[/tex] as well. Thus, we only need [tex]\( x + 3y \geq 2.5 \)[/tex].
This means the system will have solutions as long as [tex]\( x + 3y \geq 2.5 \)[/tex]. There is a region where both inequalities hold true, implying valid solutions exist.
### System 3
[tex]\[ \begin{cases} x + 3y \leq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
For this system, we have:
1. [tex]\( x + 3y \leq 0.5 \)[/tex]
2. [tex]\( x + 3y \geq 2.5 \)[/tex]
Combining these inequalities, we see that they form a contradiction:
[tex]\[ x + 3y \leq 0.5 \][/tex]
[tex]\[ x + 3y \geq 2.5 \][/tex]
Here's why it's a contradiction:
- The first inequality requires [tex]\( x + 3y \)[/tex] to be 0.5 or less.
- The second inequality requires [tex]\( x + 3y \)[/tex] to be 2.5 or more.
Both conditions cannot be true simultaneously for any [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Therefore, this system has no solutions.
### Conclusion
Based on the analysis:
- System 1 has solutions.
- System 2 has solutions.
- System 3 has no solutions.
Therefore, the system of inequalities with no solution is:
[tex]\[ \begin{cases} x + 3y \leq 0.5 \\ x + 3y \geq 2.5 \end{cases} \][/tex]
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