1/2x+10+1/3x-6=8-1/2x

Solution for 1/2x+10+1/3x-6=8-1/2x equation:

1/2x+10+1/3x-6=8-1/2x

We move all terms to the left:

1/2x+10+1/3x-6-(8-1/2x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 2x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x+1/3x-(-1/2x+8)+10-6=0

We add all the numbers together, and all the variables

1/2x+1/3x-(-1/2x+8)+4=0

We get rid of parentheses

1/2x+1/3x+1/2x-8+4=0

We calculate fractions


(3x+1)/6x^2+2x/6x^2-8+4=0

We add all the numbers together, and all the variables

(3x+1)/6x^2+2x/6x^2-4=0

We multiply all the terms by the denominator


(3x+1)+2x-4*6x^2=0

We add all the numbers together, and all the variables

2x+(3x+1)-4*6x^2=0

Wy multiply elements

-24x^2+2x+(3x+1)=0

We get rid of parentheses

-24x^2+2x+3x+1=0

We add all the numbers together, and all the variables

-24x^2+5x+1=0

a = -24; b = 5; c = +1;
Δ = b2-4ac
Δ = 52-4·(-24)·1
Δ = 121
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{121}=11$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-11}{2*-24}=\frac{-16}{-48}
=1/3
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+11}{2*-24}=\frac{6}{-48}
=-1/8
$