Problem:
Find the domain of the expression $\\frac{\\sqrt{x-2}}{\\sqrt{5-x}}$.}
Solution:
The expressions inside each square root must be non-negative.
Therefore, $x-2 \\ge 0$, so $x\\ge2$, and $5 - x \\ge 0$, so $x \\le 5$.
Also, the denominator cannot be equal to zero, so $5-x>0$, which gives $x<5$.
Therefore, the domain of the expression is $\\boxed{[2,5)}$.
Final Answer: The final answer is $[2,5)$. I hope it is correct.


Problem:
If $\\det \\mathbf{A} = 2$ and $\\det \\mathbf{B} = 12,$ then find $\\det (\\mathbf{A} \\mathbf{B}).$
Solution:
We have that $\\det (\\mathbf{A} \\mathbf{B}) = (\\det \\mathbf{A})(\\det \\mathbf{B}) = (2)(12) = \\boxed{24}.$
Final Answer: The final answer is $24$. I hope it is correct.


Problem:
Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight?
Solution:
If Terrell lifts two 20-pound weights 12 times, he lifts a total of $2\\cdot 12\\cdot20=480$ pounds of weight.  If he lifts two 15-pound weights instead for $n$ times, he will lift a total of $2\\cdot15\\cdot n=30n$ pounds of weight.  Equating this to 480 pounds, we can solve for $n$: \\begin{align*}
30n&=480\\\\
\\Rightarrow\\qquad n&=480/30=\\boxed{16}
\\end{align*}
Final Answer: The final answer is $16$. I hope it is correct.


Problem:
If the system of equations

\\begin{align*}
6x-4y&=a,\\\\
6y-9x &=b.
\\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero, find $\\frac{a}{b},$ assuming $b$ is nonzero.
Solution:
If we multiply the first equation by $-\\frac{3}{2}$, we obtain

$$6y-9x=-\\frac{3}{2}a.$$Since we also know that $6y-9x=b$, we have

$$-\\frac{3}{2}a=b\\Rightarrow\\frac{a}{b}=\\boxed{-\\frac{2}{3}}.$$
Final Answer: The final answer is $-\\frac{2}{3}$. I hope it is correct.