Let's break down the requirements given in the problem and formulate the correct inequalities step-by-step:
1. No fewer than five male students:
- This means that the number of male students, represented by [tex]\( y \)[/tex], must be at least 5. Therefore, the appropriate inequality is:
[tex]\[
y \geq 5
\][/tex]
2. Non-negativity of the number of female students:
- The number of female students, represented by [tex]\( x \)[/tex], cannot be negative. Therefore, the appropriate inequality is:
[tex]\[
x \geq 0
\][/tex]
3. Up to twelve students in total:
- The total number of students, which is the sum of female and male students ([tex]\( x + y \)[/tex]), must not exceed 12. Therefore, the appropriate inequality is:
[tex]\[
x + y \leq 12
\][/tex]
Now, let's evaluate the other inequalities given in the problem and see whether they apply to the situation:
- [tex]\( y > 5 \)[/tex]: This inequality suggests more than 5 male students, which is not required by the problem; it only needs no fewer than 5. Thus, this inequality is not appropriate.
- [tex]\( x + y < 12 \)[/tex]: This inequality suggests fewer than 12 students in total, which again is not what the problem states; it allows for up to 12 students. Thus, this inequality is not appropriate.
- [tex]\( x > 0 \)[/tex]: This inequality implies that there must be at least 1 female student, which is not a constraint given in the problem; it only states non-negativity. Thus, this inequality is not appropriate.
Based on the analysis, the correct inequalities that model the given situation are:
1. [tex]\( y \geq 5 \)[/tex]
2. [tex]\( x \geq 0 \)[/tex]
3. [tex]\( x + y \leq 12 \)[/tex]
So, the correct inequalities are:
[tex]\[ y \geq 5, x \geq 0, x + y \leq 12 \][/tex]
