1/2x-3=3/2x+4-x

Solution for 1/2x-3=3/2x+4-x equation:

1/2x-3=3/2x+4-x

We move all terms to the left:

1/2x-3-(3/2x+4-x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 2x+4-x)!=0

We move all terms containing x to the left, all other terms to the right

2x-x)!=-4

x∈R


We add all the numbers together, and all the variables

1/2x-(-1x+3/2x+4)-3=0

We get rid of parentheses

1/2x+1x-3/2x-4-3=0

We multiply all the terms by the denominator


1x*2x-4*2x-3*2x+1-3=0

We add all the numbers together, and all the variables

1x*2x-4*2x-3*2x-2=0

Wy multiply elements

2x^2-8x-6x-2=0

We add all the numbers together, and all the variables

2x^2-14x-2=0

a = 2; b = -14; c = -2;
Δ = b2-4ac
Δ = -142-4·2·(-2)
Δ = 212
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{212}=\sqrt{4*53}=\sqrt{4}*\sqrt{53}=2\sqrt{53}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{53}}{2*2}=\frac{14-2\sqrt{53}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{53}}{2*2}=\frac{14+2\sqrt{53}}{4}
$