Question

6/x-3 - 2/x^2 - 9 = 4/x+3=4/x+3

find undefined terms, denominators ldc, rewrite the equation with ldc, simplify show final equations that has no denoinator, solve for x, plug in the values for x n the original equations to check for extraneous solutions, state finals solutions

use the same steps as below:
5/(x^2-4) - 3/(x-2)
dos 5/(x^2-4) - 3/x-2
5/(x+2)(x-2) - 3/(x-2) * x+2/x+2
5-3x-6/(x-2)(x+2)
-3x-1/(x+2)(x-2)

Answer 1

Interpreting the equation as
6/(x−3) − 2/(x^2 − 9) = 4/(x+3)
(note x^2 − 9 = (x−3)(x+3)).

1) Undefined terms
x − 3 = 0 → x = 3
x + 3 = 0 → x = −3
So x ≠ 3, −3.

2) Denominators and LCM
Denominators: (x−3), (x^2 − 9) = (x−3)(x+3), (x+3).
LCM = (x−3)(x+3).

3) Multiply every term by the LCM (x−3)(x+3)
Multiply each term:

(6/(x−3))·(x−3)(x+3) = 6(x+3) = 6x + 18
(−2/(x^2 − 9))·(x−3)(x+3) = −2
(4/(x+3))·(x−3)(x+3) = 4(x−3) = 4x − 12

So the equation with no denominators is
6x + 18 − 2 = 4x − 12

4) Simplify and solve
6x + 16 = 4x − 12
6x − 4x = −12 − 16
2x = −28
x = −14

5) Check in original equation (to rule out extraneous solutions)
Plug x = −14:

LHS = 6/(−14−3) − 2/( (−14)^2 − 9 )
= 6/(−17) − 2/(196 − 9)
= −6/17 − 2/187
Convert to common denominator 187: −66/187 − 2/187 = −68/187

RHS = 4/(−14+3) = 4/(−11) = −4/11 = −68/187

LHS = RHS, and x = −14 is not excluded (−14 ≠ ±3), so it is valid.

Final solution: x = −14.