1/2 - ( x/3 - x ) = 5/3 - 1/2 x
<=> ( 1 / 2 ) - ( x / 3 ) - ( 3 x / 3 ) ) = ( 5 / 3 ) - ( 1 / 2 ) x
<=> ( 1 / 2 ) - ( - 2 x / 3 ) = ( 5 / 3 ) - ( 1 / 2 ) x
<=> ( 1 / 2 ) + ( 2 x / 3 ) = ( 5 / 3 ) - ( x / 2 )
<=> ( 2 x / 3 )+ ( x / 2 ) = ( 5 / 3 ) - ( 1 / 2 )
<=> ( 4 x / 6 ) + ( 3 x / 6 ) = ( 10 / 6 ) - ( 3 / 6 )
<=> 7 x / 6 = 7 / 6
<=> 7 x = 7
<=> x = 1
Zweite Aufgabe:
4/5 x - ( 1 - 4/3 x ) = 4/5 + 1/3
<=> ( 4 x / 5 ) - ( 1 - ( 4 / 3 ) x ) = ( 4 / 5 ) + ( 1 / 3 )
<=> ( 4 x / 5 ) - 1 + ( 4 x / 3 ) = ( 12 / 15 ) + ( 5 / 15 )
<=> ( 4 x / 5 ) + ( 4 x / 3 ) ) = ( 17 / 15 ) + 1
<=> ( 12 x / 15 ) + ( 20 x / 15 ) = ( 17 / 15 ) + ( 15 / 15 )
<=> 32 x / 15 = 32 / 15
<=> 32 x = 32
<=> x = 1
Hier das Ganze in TeX:
Erste Aufgabe:
$$\frac { 1 }{ 2 } -\left( \frac { x }{ 3 } -x \right) =\frac { 5 }{ 3 } -\frac { 1 }{ 2 } x$$$$\Leftrightarrow \frac { 1 }{ 2 } -\left( \frac { x }{ 3 } -\frac { 3x }{ 3 } \right) =\frac { 5 }{ 3 } -\frac { x }{ 2 }$$$$\Leftrightarrow \frac { 1 }{ 2 } -\left( -\frac { 2x }{ 3 } \right) =\frac { 5 }{ 3 } -\frac { x }{ 2 }$$$$\Leftrightarrow \frac { 1 }{ 2 } +\frac { 2x\quad }{ 3 } =\frac { 5 }{ 3 } -\frac { x }{ 2 }$$$$\Leftrightarrow \frac { 2x }{ 3 } +\frac { x }{ 2 } =\frac { 5 }{ 3 } -\frac { 1 }{ 2 }$$$$\Leftrightarrow \frac { 4x }{ 6 } +\frac { 3x }{ 6 } =\frac { 10 }{ 6 } -\frac { 3 }{ 6 }$$$$\Leftrightarrow \frac { 7x }{ 6 } =\frac { 7 }{ 6 }$$$$\Leftrightarrow 7x=7$$$$\Leftrightarrow x=1$$
Zweite Aufgabe:
$$\frac { 4 }{ 5 } x-\left( 1-\frac { 4 }{ 3 } x \right) =\frac { 4 }{ 5 } +\frac { 1 }{ 3 }$$$$\Leftrightarrow \frac { 4x }{ 5 } -1+\frac { 4x }{ 3 } =\frac { 12 }{ 15 } +\frac { 5 }{ 15 }$$$$\Leftrightarrow \frac { 12x }{ 15 } +\frac { 20x }{ 15 } =\frac { 12 }{ 15 } +\frac { 5 }{ 15 } +1$$$$\Leftrightarrow \frac { 32x }{ 15 } =\frac { 12 }{ 15 } +\frac { 5 }{ 15 } +\frac { 15 }{ 15 }$$$$\Leftrightarrow \frac { 32x }{ 15 } =\frac { 32 }{ 15 }$$$$\Leftrightarrow 32x=32$$$$\Leftrightarrow x=1$$
