How do you solve #(x-1)/(x-2) - (x+1)/(x+2) = 4/(x^2-4)#?

Question

Julia Adams · Accepted Answer

There are no valid solutions to this equation

Note that #(x-1)/(x-2)-(x+1)/(x+2)=4/(x^2-4)# is only defined if #x!=+-2# If we attempt to solve this equation by converting all terms to the common denominator of #(x-2)(x+2)=x^2-4# we get #((x-1)(x+2))/(x^2-4)-((x+1)(x-2))/(x^2-4)=4/(x^2-4)# #rarr (x-1)(x+2)-(x+1)(x-2)=4# #rarr (x^2+x-2)-(x^2-x-2)=4# #rarr cancel(x^2)+xcancel(-2)cancel(-x^2)+xcancel(+2)=4# #rarr 2x=4# #rarr x=2# BUT the original equation is not defined if #x=2# Consequently, there isn't a workable answer.

Eli Assouline · Answer

undefined To solve the equation (x-1)/(x-2) - (x+1)/(x+2) = 4/(x^2-4), we can follow these steps:

1. Start by finding a common denominator for all the fractions involved. In this case, the common denominator is (x-2)(x+2).

2. Multiply each term by the common denominator to eliminate the fractions.

3. Simplify the equation by distributing and combining like terms.

4. Continue simplifying until you have a quadratic equation.

5. Solve the quadratic equation by factoring or using the quadratic formula.

6. Check for any extraneous solutions by plugging them back into the original equation.

By following these steps, you should be able to find the solution(s) to the given equation.