Translating a word problem (hard)

Question

2 divided by the difference of a number and 3 minus 4 divided by the number plus 3, equals 8 times the reciprocal of the difference of a number squared and 9. What is the number. I know the answer is 5.

Answer 1

2/(x-3) - 4/(x+3) = 8(1/(x²-9)

substitute 5 for x

2/2 - 4/8 = 8(1/16)

1-1/2 =1/2

1/2 = 1/2

5 is a correct answer

factor the denominator on the right side of the original equation

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

combine the left side with a common denominator

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

or (2(x+3) - 4(x-3))/(x-3)(x+3) = 8/(x-3)(x+3)

or (2x+6-4x+12)/(x-3)(x+3)=8/(x-3)(x+3)

or (-2x+18)/(x-3)(x+3)=8/(x-3)(x+3)

or -2x+18 = 8

-2x = -10

x=-10/-2 = 5

But there's a squared term in the original equation suggesting multiple solutions.

If there are multiple solutions they may be repeated or imaginary.

Multiplying by a factor including the unknown variable also may introduce false solutions, but not here.