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 \boxed{$[2,5)$}. 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 \boxed{$24$}. 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 \boxed{$16$}. 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 \boxed{$-\frac{2}{3}$}. I hope it is correct.

Problem:
