2 < | x + 9 x − 6 | < 4 {\displaystyle {\begin{aligned}&2<\left|{\frac {x+9}{x-6}}\right\vert <4\\\end{aligned}}}
2 < | x + 9 x − 6 | {\displaystyle {\begin{aligned}2<\left|{\frac {x+9}{x-6}}\right\vert \end{aligned}}}
וגם
| x + 9 x − 6 | < 4 {\displaystyle {\begin{aligned}\left|{\frac {x+9}{x-6}}\right\vert <4\end{aligned}}}
x + 9 x − 6 > 2 x + 9 x − 6 − 2 > 0 x + 9 − 2 ( x − − 6 ) x − 6 > 0 x + 9 − 2 x + 12 x − 6 > 0 − x + 21 x − 6 > 0 / ∗ ( x − 6 ) 2 ( − x + 21 ) ( x − 6 ) > 0 ( − x + 21 ) ( x − 6 ) = 0 x 1 = 21 x 2 = 6 ↓ 6 < x < 21 {\displaystyle {\begin{aligned}&{\frac {x+9}{x-6}}>2&{\frac {x+9}{x-6}}-2>0\\&{\frac {x+9-2(x--6)}{x-6}}>0\\&{\frac {x+9-2x+12}{x-6}}>0\\&{\frac {-x+21}{x-6}}>0/*(x-6)^{2}\\&(-x+21)(x-6)>0\\&(-x+21)(x-6)=0\\&x_{1}=21&x_{2}=6\\&\downarrow \\&6<x<21\end{aligned}}}
ואו
x + 9 x − 6 < − 2 x + 9 + 2 ( x − 6 ) x − 6 < 0 x + 9 + 2 x − 12 x − 6 < 0 3 x − 3 x − 6 < 0 / ∗ ( x − 6 ) 2 ( 3 x − 3 ) ( x − 6 ) < 0 ( 3 x − 3 ) ( x − 6 ) = 0 x 1 = 1 x 2 = 6 ↓ 1 < x < 6 {\displaystyle {\begin{aligned}&{\frac {x+9}{x-6}}<-2\\&{\frac {x+9+2(x-6)}{x-6}}<0\\&{\frac {x+9+2x-12}{x-6}}<0\\&{\frac {3x-3}{x-6}}<0/*(x-6)^{2}\\&(3x-3)(x-6)<0\\&(3x-3)(x-6)=0\\&x_{1}=1&x_{2}=6&\downarrow \\&1<x<6\end{aligned}}}
x + 9 x − 6 > − 4 x + 9 x − 6 + 4 > 0 x + 9 + 4 ( x − 6 ) x − 6 > 0 x + 9 + 4 x − 24 x − 6 > 0 5 x − 15 x − 6 > 0 / ∗ ( x + 6 ) 2 ( 5 x − 15 ) ( x − 6 ) > 0 ( 5 x − 15 ) ( x − 6 ) = 0 x 1 = 3 x 2 = 6 ↓ x < 3 ; x > 6 {\displaystyle {\begin{aligned}&{\frac {x+9}{x-6}}>-4\\&{\frac {x+9}{x-6}}+4>0\\&{\frac {x+9+4(x-6)}{x-6}}>0\\&{\frac {x+9+4x-24}{x-6}}>0\\&{\frac {5x-15}{x-6}}>0/*(x+6)^{2}\\&(5x-15)(x-6)>0\\&(5x-15)(x-6)=0\\&x_{1}=3&x_{2}=6\\&\downarrow \\&x<3;x>6\end{aligned}}}
x + 9 x − 6 < 4 x + 9 − 4 ( x − 6 ) x − 6 < 0 x + 9 − 4 x + 24 x − 6 − 3 x + 33 x − 6 < 0 / ∗ ( x − 6 ) 2 ( − 3 x + 33 ) ( x − 6 ) < 0 ( − 3 x + 33 ) ( x − 6 ) = 0 x 1 = 11 x 2 = 6 ↓ x > 11 ; x < 6 {\displaystyle {\begin{aligned}&{\frac {x+9}{x-6}}<4\\&{\frac {x+9-4(x-6)}{x-6}}<0\\&{\frac {x+9-4x+24}{x-6}}\\&{\frac {-3x+33}{x-6}}<0/*(x-6)^{2}\\&(-3x+33)(x-6)<0\\&(-3x+33)(x-6)=0\\&x_{1}=11&x_{2}=6\\&\downarrow \\&x>11;x<6\end{aligned}}}
1 < x < 21 {\displaystyle \ 1<x<21}
x < 3 ; x > 11 {\displaystyle \ x<3;x>11}
1 < x < 3 ; 11 < x < 21 {\displaystyle \ 1<x<3;11<x<21}
