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

### Question Description

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)) = (9x-9x) / (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)

(24-24)/(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)

(24-24)/(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 Duke University Anonymous
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