Đáp án:

$\begin{array}{l}
1)a)\dfrac{{3x - 5}}{{x + 4}}\\
 = \dfrac{{3.x + 12 - 17}}{{x + 4}}\\
 = \dfrac{{3\left( {x + 4} \right) - 17}}{{x + 4}}\\
\dfrac{{3x - 5}}{{x + 4}} \in Z\\
 \Rightarrow \left( {3x - 5} \right) \vdots \left( {x + 4} \right)\\
 \Rightarrow \left( {3\left( {x + 4} \right) - 17} \right) \vdots \left( {x + 4} \right)\\
 \Rightarrow 17 \vdots \left( {x + 4} \right)\left( {do:3.\left( {x + 4} \right) \vdots \left( {x + 4} \right)} \right)\\
 \Rightarrow \left( {x + 4} \right) \in \left\{ { - 17; - 1;1;17} \right\}\\
 \Rightarrow x \in \left\{ { - 21; - 5; - 3;13} \right\}\\
2)a)\dfrac{a}{b} = \dfrac{{ - 3}}{{11}}\\
 \Rightarrow \dfrac{a}{{ - 3}} = \dfrac{b}{{11}} = k \Rightarrow \left\{ \begin{array}{l}
a =  - 3.k\\
b = 11.k
\end{array} \right.\\
Do:a + b = 40\\
 \Rightarrow  - 3.k + 11.k = 40\\
 \Rightarrow 8.k = 40\\
 \Rightarrow k = 5\\
 \Rightarrow \left\{ \begin{array}{l}
a =  - 3.k =  - 15\\
b = 11.k = 55
\end{array} \right.\\
 \Rightarrow \dfrac{a}{b} = \dfrac{{ - 15}}{{55}}\\
b)\dfrac{a}{b} = \dfrac{{ - 13}}{{19}}\\
 \Rightarrow \left\{ \begin{array}{l}
a =  - 13k\\
b = 19.k
\end{array} \right.\\
a + b = 18\\
 \Rightarrow  - 13.k + 19.k = 18\\
 \Rightarrow 6.k = 18\\
 \Rightarrow k = 3\\
 \Rightarrow a =  - 39;b = 57\\
 \Rightarrow \dfrac{a}{b} = \dfrac{{ - 39}}{{57}}\\
c)\dfrac{a}{b} = \dfrac{{17}}{{23}}\\
 \Rightarrow \left\{ \begin{array}{l}
a = 17k\\
b = 23k
\end{array} \right.\\
b - a = 30\\
 \Rightarrow 23k - 17k = 30\\
 \Rightarrow 6k = 30\\
 \Rightarrow k = 5\\
 \Rightarrow \left\{ \begin{array}{l}
a = 85\\
b = 115
\end{array} \right.\\
 \Rightarrow \dfrac{a}{b} = \dfrac{{85}}{{115}}
\end{array}$

Tham khảo

 `1)` Đặt `A=\frac{3x-5}{x+4}`

Để `A` là số nguyên `⇔3x-5 \vdots x+4`

Xét hiệu:

`⇒(3x-5)-3(x+4) \vdots x+4`

`⇒3x-5-3x-12 \vdots x+4`

`⇒-17 \vdots x+4`

`⇒x+4∈Ư(1,-1,17,-17)`

`⇒x∈{-3,-5,13,-21}`

Câu `2:`

`a)` Vì `\frac{a}{b}=\frac{-3}{11}`

`⇒a=\frac{-3}{11}b`

Do đó `a+b=40`

`⇒\frac{-3}{11}b+b=40`

`⇒b.(\frac{-3}{11}+1)=40`

`⇒b.\frac{8}{11}=40`

`⇒b=40:\frac{8}{11}`

`⇒b=55`

`⇒a=55.\frac{-3}{11}=-15`

Vậy `\frac{a}{b}=\frac{-15}{55}`

`b)` Vì `\frac{a}{b}=\frac{-13}{19}`

`⇒a=\frac{-13}{19}b`

Do đó `a+b=18`

`⇒\frac{-13}{19}b+b=18`

`⇒b.(\frac{-13}{19}+1)=18`

`⇒b.\frac{6}{19}=18`

`⇒b=18:\frac{6}{19}`

`⇒b=57`

`⇒a=57.\frac{-13}{19}=-39`

Vậy `\frac{a}{b}=\frac{-39}{57}`

`c)` Tự làm 

Đáp án:`\frac{a}{b}=\frac{85}{115}`

$\text{©CBT}$