Certainly! Let's analyze the given inequalities step by step to ensure they are represented correctly and identify any errors if present.
### Inequality A
Inequality A is given as:
[tex]\[ y \leq -\frac{11}{3} x - 8 \][/tex]
### Inequality B
Inequality B is given as:
[tex]\[ y \geq 3 \][/tex]
### Checking Specific Points
To verify these inequalities, let's check a specific point and see if it satisfies both inequalities.
#### For [tex]\(x = 0\)[/tex]:
1. Inequality A:
Substituting [tex]\(x = 0\)[/tex] into Inequality A:
[tex]\[ y \leq -\frac{11}{3} \cdot 0 - 8 \][/tex]
[tex]\[ y \leq -8 \][/tex]
So, when [tex]\(x = 0\)[/tex], Inequality A simplifies to [tex]\( y \leq -8 \)[/tex].
2. Inequality B:
Inequality B is:
[tex]\[ y \geq 3 \][/tex]
### Conclusion
Now we have two conditions for the point [tex]\(x = 0\)[/tex]:
1. From Inequality A: [tex]\( y \leq -8 \)[/tex]
2. From Inequality B: [tex]\( y \geq 3 \)[/tex]
Clearly, these two inequalities are inconsistent when evaluated at [tex]\(x = 0\)[/tex] because [tex]\( y \)[/tex] cannot be simultaneously less than or equal to [tex]\(-8\)[/tex] and greater than or equal to [tex]\(3\)[/tex].
### Verification of Correct Inequalities
1. Inequality A:
When substituting [tex]\(x = 0\)[/tex], the result is [tex]\( y \leq -8 \)[/tex].
[tex]\[ -8 \leq -8 \][/tex]
This is \textbf{true}, so the representation of Inequality A is correct.
2. Inequality B:
The Inequality remains [tex]\( y \geq 3 \)[/tex].
[tex]\[ 3 \geq 3 \][/tex]
This is also \textbf{true}, so the representation of Inequality B is correct.
### Summary
Both individual inequalities are correctly represented:
- Inequality A: [tex]\( y \leq -\frac{11}{3} x - 8 \)[/tex] is correctly formed.
- Inequality B: [tex]\( y \geq 3 \)[/tex] is correctly formed.
However, it's important to note that the conditions defined by these inequalities do not overlap for any real number [tex]\(x\)[/tex] because they present mutually exclusive conditions for [tex]\(y\)[/tex]. Thus, no common solutions exist within the intersection of these two inequalities.
In conclusion, there's no error in the representation of the inequalities themselves. The only observation is the non-overlapping nature of their solutions.
