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a) Ta có: \(M=\frac{2x+5}{x+1}=\frac{2\left(x+1\right)+3}{x+1}=\frac{2x+2+3}{x+1}\)
Vì \(2x+2⋮\left(x+1\right)\Rightarrow3⋮\left(x+1\right)\)
Nên \(x+1\inƯ\left(3\right)=\left\{1;-1;3;-3\right\}\)
\(\Rightarrow x=\left\{0;-2;2;-4\right\}\)
b) Tương tự
1,b, 2xy - x = y + 5
<=> 4xy - 2x = 2y + 10
<=> 2x(2y - 1) - (2y - 1) = 11
<=> (2x - 1)(2y - 1) = 11
Lập bảng ra làm nốt
\(1,c,\frac{1}{x}-3=-\frac{1}{y-2}\)
\(\Leftrightarrow y-2-3x\left(y-2\right)=-x\)
\(\Leftrightarrow y-2-3xy+6x+x=0\)
\(\Leftrightarrow-3xy+7x+y-2=0\)
\(\Leftrightarrow-x\left(3y-7\right)+y-2=0\)
\(\Leftrightarrow-3x\left(3y-7\right)+3y-6=0\)
\(\Leftrightarrow-3x\left(3y-7\right)+\left(3y-7\right)=-1\)
\(\Leftrightarrow\left(1-3x\right)\left(3y-7\right)=-1\)
Lập bảng làm nốt
\(D=\frac{4x+1}{x+3}\inℤ\Leftrightarrow4x+1⋮x+3\)
\(\Rightarrow4x+12-11⋮x+3\)
\(\Rightarrow4\left(x+3\right)-11⋮x+3\)
\(\Rightarrow11⋮x+3\)
\(\Rightarrow x+3\in\left\{-1;1;-11;11\right\}\)
\(\Rightarrow x\in\left\{-4;-2;-14;8\right\}\)
a) \(D=\frac{4x+1}{x+3}\)
=> 4x + 1 \(⋮\)( x + 3 ) để D là số nguyên
Mà ( x + 3 ) \(⋮\)( x + 3 ) => 4( x + 3 ) \(⋮\)( x + 3 )
=> [ 4x + 1 - 4( x + 3 ) ] \(⋮\)( x + 3 )
=> [ 4x + 1 - 4x + 12 ] \(⋮\)( x + 3 )
=> 13 \(⋮\)( x + 3 )
=> \(x+3\inƯ\left(13\right)\)\(=\left\{\pm1;\pm13\right\}\)
| x + 3 | -1 | 1 | -13 | 13 |
| x | 2 | 4 | -10 | 16 |
Vậy \(x\in\left\{-10;2;4;16\right\}\)Để D là số nguyên
b) \(E=\frac{6x+2}{2x-3}\)
=> 6x + 2 \(⋮\)2x - 3 để E là số nguyên
Mà ( 2x - 3 ) \(⋮\)( 2x - 3 ) => 3( 2x - 3 ) \(⋮\)( 2x - 3 )
=> [ 6x + 2 - 3( 2x - 3 ) ] \(⋮\)( 2x - 3 )
=> [ 6x + 2 - 6x - 3 ] \(⋮\)( 2x - 3 )
=> -1 \(⋮\)( 2x - 3 )
=> ( 2x - 3 ) \(\inƯ\left(-1\right)=\left\{\pm1\right\}\)
| 2x - 3 | -1 | 1 |
| 2x | 2 | 4 |
| x | 1 | 2 |
Vậy x \(\in\left\{1;2\right\}\)để E là số nguyên
Còn phần còn lại cậu có thể làm tương tự.
1.b) \(\left(\left|x\right|-3\right)\left(x^2+4\right)< 0\)
\(\Rightarrow\hept{\begin{cases}\left|x\right|-3\\x^2+4\end{cases}}\) trái dấu
\(TH1:\hept{\begin{cases}\left|x\right|-3< 0\\x^2+4>0\end{cases}}\Leftrightarrow\hept{\begin{cases}\left|x\right|< 3\\x^2>-4\end{cases}}\Leftrightarrow x\in\left\{0;\pm1;\pm2\right\}\)
\(TH1:\hept{\begin{cases}\left|x\right|-3>0\\x^2+4< 0\end{cases}}\Leftrightarrow\hept{\begin{cases}\left|x\right|>3\\x^2< -4\end{cases}}\Leftrightarrow x\in\left\{\varnothing\right\}\)
Vậy \(x\in\left\{0;\pm1;\pm2\right\}\)
\(\frac{1}{3}-|\frac{5}{4}-2x|=\frac{1}{4}\)
\(\Leftrightarrow|\frac{5}{4}-2x|=\frac{1}{4}+\frac{1}{3}=\frac{7}{12}\)
\(\Leftrightarrow\orbr{\begin{cases}Th1:\frac{5}{4}-2x=\frac{7}{12}\\Th2:\frac{5}{4}-2x=-\frac{7}{12}\end{cases}}\)
\(\Leftrightarrow Th1:\frac{5}{4}-2x=\frac{7}{12}\) \(\Leftrightarrow Th2:\frac{5}{4}-2x=-\frac{7}{12}\)
\(\Leftrightarrow2x=\frac{7}{12}+\frac{5}{4}\) \(\Leftrightarrow2x=-\frac{7}{12}+\frac{5}{4}\)
\(\Leftrightarrow2x=\frac{11}{6}\) \(\Leftrightarrow2x=\frac{2}{3}\)
\(\Leftrightarrow x=\frac{11}{12}\) \(\Leftrightarrow x=\frac{1}{3}\)
P/s : Mình làm bừa ạ nếu kh đúng xin mọi người chỉ thêm ~~
b)\(B=\frac{x^2-3x+7}{x-3}=\frac{x\left(x-3\right)+7}{x-3}=x+\frac{7}{x-3}\)
\(\Rightarrow B\in Z\Leftrightarrow x+\frac{7}{x-3}\in Z\Leftrightarrow x\in Z,\frac{7}{x-3}\in Z\Leftrightarrow7⋮x-3\Leftrightarrow x-3\inƯ\left\{7\right\}\)
\(\Rightarrow x-3\in\left\{-1;-7;1;7\right\}\)
\(\Rightarrow x\in\left\{2;-4;4;10\right\}\)
c)\(C=\frac{x^2+1}{x-1}=\frac{x^2-1+2}{x-1}=\frac{\left(x-1\right)\left(x+1\right)+2}{x-1}=\left(x+1\right)+\frac{2}{x-1}\)
\(\Rightarrow C\in Z\Leftrightarrow\left(x+1\right)+\frac{2}{x-1}\in Z\Leftrightarrow x-1\in Z;\frac{2}{x-1}\in Z\)
\(\Leftrightarrow x\in Z;2⋮x-1\Rightarrow x-1\inƯ\left(2\right)\)
\(\Rightarrow x-1\in\left\{-1;-2;1;2\right\}\)
\(\Rightarrow x\in\left\{0;-1;2;3\right\}\)
\(a,\frac{-24}{x}+\frac{18}{x}=\frac{-24+18}{x}=\frac{-6}{x}\)
\(\Leftrightarrow x\inƯ(-6)=\left\{\pm1;\pm2;\pm3;\pm6\right\}\)
\(b,\frac{2x-5}{x+1}=\frac{2x+2-7}{x+1}=\frac{2(x+1)-7}{x+1}=2-\frac{7}{x+1}\)
\(\Leftrightarrow7⋮x+1\Leftrightarrow x+1\inƯ(7)=\left\{\pm1;\pm7\right\}\)
Xét các trường hợp rồi tìm được x thôi :>
\(c,\frac{3x+2}{x-1}-\frac{x-5}{x-1}=\frac{3x+2-x-5}{x-1}=\frac{2x+7}{x-1}=\frac{2x-2+9}{x-1}=\frac{2(x-1)+9}{x-1}=2+\frac{9}{x-1}\)
\(\Leftrightarrow9⋮x-1\Leftrightarrow x-1\inƯ(9)=\left\{\pm1;\pm3;\pm9\right\}\)
\(\Leftrightarrow x\in\left\{2;0;4;-2;10;-8\right\}\)
d, TT
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