Question

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

Answer 1

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

1) Undefined terms (values that make any denominator 0):
- x − 3 = 0 → x = 3
- x^2 − 9 = 0 → x = ±3
- x + 3 = 0 → x = −3
So x ≠ 3 and x ≠ −3.

2) Denominators and LCD:
Denominators: x − 3, x^2 − 9, x + 3.
Factor x^2 − 9 = (x − 3)(x + 3). The least common denominator (LCD) is x^2 − 9 = (x − 3)(x + 3).

3) Multiply the entire equation by the LCD to clear denominators:
(x^2 − 9)·[6/(x − 3) − 2/(x^2 − 9)] = (x^2 − 9)·[4/(x + 3)].

Simplify each term:
(x^2 − 9)·6/(x − 3) = 6(x + 3)
(x^2 − 9)·[−2/(x^2 − 9)] = −2
(x^2 − 9)·4/(x + 3) = 4(x − 3)

So the equation with no denominators is:
6(x + 3) − 2 = 4(x − 3).

4) Solve the resulting equation:
6x + 18 − 2 = 4x − 12
6x + 16 = 4x − 12
2x = −28
x = −14.

5) Check x = −14 in the original equation:
Left: 6/(−14 − 3) − 2/( (−14)^2 − 9 ) = 6/(−17) − 2/(196 − 9) = −6/17 − 2/187
Convert to common denominator 187: −6/17 = −66/187, so left = −66/187 − 2/187 = −68/187 = −4/11.
Right: 4/(−14 + 3) = 4/(−11) = −4/11.
Left = Right, so x = −14 is valid and not excluded by the domain.

Final solution: x = −14 (with x ≠ 3, −3).