Identify the Value(s) of the Variable that Makes It Undefined
Mathematics

Tutor: None Selected  Time limit: 1 Day 
Write the value or values of the variable that makes the denominator zero. Then solve the equation.
3/x = 9/8x + 3.
I am not sure about the equation, so will solve 3 options.
Option 1: 3/x = 9/(8x + 3)
On the left hand side x=0 will make denominator zero.
On the right hand side denominator equal zero will mean
8x + 3 = 0 or 8x + 3  3 = 0  3 or 8x = 3 or x = 3/8
So x=0 and x=3/8 make denominator (one of them at a time) equal to zero.
To solve this, make it to common denominator
3(8x+3) / (x(8x+3)) = 9x / (x(8x+3))
Move everything to one side
( 3(8x+3)  9x ) / (x(8x+3)) = (9x9x) / (x(8x+3))
Simplify
(24x + 9  9x) / (x(8x+3)) = 0
(15x + 9) / (x(8x+3)) =0
The fraction can be zero only if the numenator is zero
15x + 9 = 0 or 15x = 9 or x = 9/15 = 3/5 = 0.6
Option 2: 3/x = (9/(8x)) + 3
On the left hand side x=0 will make denominator zero.
On the right hand side 8x=0 will make denominator zero, which also possible only when x=0.
To solve this, make it to common denominator
3*8 / (8x) = (9/(8x)) + 3*(8x)/(8x) or 24 / (8x) = (9/(8x)) + (24x/(8x))
Move everything to one side (right instead of left as in previous case)(2424)/(8x) = (9 + 24x 24)/(8x) or 0 = (24x  15)/(8x)
The fraction can be zero only if the numenator is zero
24x  15 = 0 or 24x = 15 or x = 15/24 = 5/8 = 0.625
Option 3: 3/x = x*(9/8) + 3
The only denominator here is on the left hand side of equation and to make it zero means x=0
To solve this let us get it to common denominator
3*8/(8x) = (9*x*x)/(8x) + (3*8x)/(8x) or 24/(8x) = (9x^2)/(8x) + (24x)/(8x)
Move everything to one side (right hand side)
(2424)/(8x) = (9x^2 + 24x 24)/(8x) or 0 = (9x^2 + 24x 24)/(8x)
The fraction can be zero only if the numenator is zero
9x^2 + 24x  24 =0 or (divide by 3) 3x^2 +8x  8 = 0
To solve quadratic equation let us find discriminant:
D=8^2  4*3*(8)=64 + 96 = 160 = (4 * root(10))^2
The solutions are
x = (8 + 4 * root(10))/(2*3) = (4 + 2 * root(10)) / 3
and
x = (8  4 * root(10))/(2*3) = (4  2 * root(10)) / 3
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