\(\dfrac{1}{2}+\dfrac{-1}{3}+\dfrac{-2}{3}\le x< \dfrac{-3}{5}+\dfrac{1}{6}+\dfrac{-2}{5}+\dfrac{3}{2}\)

\(\Leftrightarrow\dfrac{1}{2}+\left(\dfrac{-1}{3}+\dfrac{-2}{3}\right)\le x< \left(\dfrac{-3}{5}+\dfrac{-2}{5}\right)+\left(\dfrac{1}{6}+\dfrac{3}{2}\right)\)

\(\Leftrightarrow\dfrac{1}{2}+\left(-1\right)\le x< -1+\dfrac{5}{3}\)

\(\Leftrightarrow\dfrac{-1}{2}\le x< \dfrac{2}{3}\)

\(\Leftrightarrow\dfrac{-3}{6}\le x< \dfrac{4}{6}\)

\(\Leftrightarrow x\in\left\{-3;-2;-1;0;1;2;3\right\}\)

⇔x∈{−3;−2;−1;0;1;2;3}

\(\dfrac{5}{x}+\dfrac{y}{4}=\dfrac{1}{8}\left(x;y\in Z\right)\)

\(MSC:8x\left(x\ne0\right)\)

\(pt\Leftrightarrow\dfrac{40+2xy}{8x}=\dfrac{x}{8x}\)

\(\Leftrightarrow40+2xy=x\)

\(\Leftrightarrow x-2xy=40\)

\(\Leftrightarrow x\left(1-2y\right)=40\)

\(\Leftrightarrow x;\left(1-2y\right)\in U\left(40\right)=\left\{-1;1;-2;2;-4;4;-5;5;-8;8;-10;10;-20;20;-40;40\right\}\)

Bạn lập bảng sẽ tìm ra các cặp \(\left(x;y\in Z\right)\) nhé!

a) Để y nguyên thì \(6x-4⋮2x+3\)

\(\Leftrightarrow-13⋮2x+3\)

\(\Leftrightarrow2x+3\in\left\{1;-1;13;-13\right\}\)

\(\Leftrightarrow2x\in\left\{-2;-4;10;-16\right\}\)

hay \(x\in\left\{-1;-2;5;-8\right\}\)

b:

ĐKXĐ: x<>0

 \(\dfrac{2}{x}+\dfrac{y}{3}=\dfrac{1}{6}\)

=>\(\dfrac{6+xy}{3x}=\dfrac{1}{6}\)

=>\(6\left(6+xy\right)=3x\)

=>\(x=2\left(6+xy\right)=12+2xy\)

=>\(x\left(1-2y\right)=12\)

mà x,y là các số nguyên

nên \(\left(x;1-2y\right)\in\left\{\left(12;1\right);\left(-12;-1\right);\left(4;3\right);\left(-4;-3\right)\right\}\)

=>\(\left(x,y\right)\in\left\{\left(12;0\right);\left(-12;1\right);\left(4;-1\right);\left(-4;2\right)\right\}\)

c: ĐKXĐ: y<>-1

\(\dfrac{x}{3}+\dfrac{1}{y+1}=\dfrac{1}{6}\)

=>\(\dfrac{xy+x+3}{3\left(y+1\right)}=\dfrac{1}{6}\)

=>\(\dfrac{2\left(xy+x+3\right)}{6\left(y+1\right)}=\dfrac{y+1}{6\left(y+1\right)}\)

=>\(2xy+2x+6=y+1\)

=>\(2x\left(y+1\right)-\left(y+1\right)=-6\)

=>\(\left(2x-1\right)\left(y+1\right)=-6\)

mà x,y là các số nguyên

nên \(\left(2x-1;y+1\right)\in\left\{\left(1;-6\right);\left(-1;6\right);\left(3;-2\right);\left(-3;2\right)\right\}\)

=>\(\left(x,y\right)\in\left\{\left(1;-7\right);\left(0;5\right);\left(2;-3\right);\left(-1;1\right)\right\}\)

Bài 3 :

\(\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+...+\dfrac{1}{2023!}\)

\(\dfrac{1}{2!}=\dfrac{1}{2.1}=1-\dfrac{1}{2}< 1\)

\(\dfrac{1}{3!}=\dfrac{1}{3.2.1}=1-\dfrac{1}{2}-\dfrac{1}{3}< 1\)

\(\dfrac{1}{4!}=\dfrac{1}{4.3.2.1}< \dfrac{1}{3!}< \dfrac{1}{2!}< 1\)

.....

\(\)\(\dfrac{1}{2023!}=\dfrac{1}{2023.2022....2.1}< \dfrac{1}{2022!}< ...< \dfrac{1}{2!}< 1\)

\(\Rightarrow\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+...+\dfrac{1}{2023!}< 1\)

Bạn xem lại đề 2, phần mẫu của N

C