Algebraic fractions, unable to solve?

Question

Without the use of a calculator, solve:

 

  2  =  x   -3

x-2    x-3

 

Previous questions have tended to rely on quadratics, but I couldn't solve it this way when I tried myself.

Answer is x= 3/2 and 4, but how do I find this?

 

Thank you in advance for any assistance.

Edward

Answer 1

  2             x

------  =  ------  -  3

(x-2)        (x-3)

 

First, get rid of the denominators by multiplying both sides of the equation by (x-3)(x-2), which gives you:

 

2(x-3) = x(x-2) - 3(x-3)(x-2)

 

When you multiply that out and combine like terms, it reduces to:

 

2x² - 11x + 12 = 0

 

Which factors to:

 

(2x-3)(x-4) = 0

 

x = 3/2 and 4