Giải
\(\left(3x+1\right)⋮\left(2x+3\right)\)
\(\Leftrightarrow\left[4\left(3x+1\right)\right]⋮\left(2x+3\right)\)
\(\Leftrightarrow\left[12x+4\right]⋮\left(2x+3\right)\)
\(\Leftrightarrow\left(12x+18-14\right)⋮\left(2x+3\right)\)
\(\Leftrightarrow\left[6\left(2x+3\right)-14\right]⋮\left(2x+3\right)\)
Vì \(\left[6\left(2x+3\right)\right]⋮\left(2x+3\right)\) nên \(14⋮\left(2x+3\right)\)
\(\Leftrightarrow2x+3\inƯ\left(14\right)=\left\{\pm1;\pm2;\pm7;\pm14\right\}\)
Mà 2x + 3 là số lẻ nên \(2x+3\in\left\{\pm1;\pm7\right\}\)
Ta có bảng sau :
| 2x + 3 | -1 | 1 | 17 | -17 |
| x | -2 | -1 | 7 | -10 |
Vậy x \(\in\) { -2 ; -1 ; 7 ; -10 }
Có \(\left(3x+1\right)⋮\left(2x+3\right)\)
\(\Rightarrow6x+2⋮2x+3\)
\(\Rightarrow6x+9-7⋮2x+3\)
\(\Rightarrow7⋮2x+3\)
\(\Rightarrow2x+3\inƯ\left(7\right)=\left\{\pm1;\pm7\right\}\)
Với 2x + 3 = 1 => x = -1 (tm)
Với 2x + 3 = -1 => x = -2 (tm)
Với 2x + 3 = 7 => x = 2 (tm)
Với 2x + 3 = -7 => x = -5 (tm)
Vậy...
\(\left(3x+1\right)⋮\left(2x+3\right)\)
\(=>\left(2x+3\right)-\left(3x+1\right)⋮2x+3\)\(=>3\left(2x+3\right)-2\left(3x+1\right)⋮2x+3\)
\(=>\left(6x+9\right)-\left(6x+2\right)⋮2x+3\)\(=>7⋮2x+3\)
\(=>2x+3\inƯ\left(7\right)\)\(=>2x+3\in\left\{\pm1;\pm7\right\}\)
\(=>X=\left\{-2;-1;-5;2\right\}\)