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

Answer 1

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

1) Undefined terms (where denominators = 0)
- x − 3 = 0 → x = 3
- x + 3 = 0 → x = −3
So x ≠ 3 and x ≠ −3.

2) Denominators and least common denominator (LCD)
- Denominators: x − 3, x^2 − 9, x + 3.
- Note x^2 − 9 = (x − 3)(x + 3).
- LCD = (x − 3)(x + 3) = x^2 − 9.

3) Multiply both sides by the LCD to clear denominators
Multiply the whole equation by (x − 3)(x + 3):

(x − 3)(x + 3) * [6/(x − 3) − 2/(x^2 − 9)] = (x − 3)(x + 3) * [4/(x + 3)]

Simplify each term:
- 6/(x − 3) * (x − 3)(x + 3) = 6(x + 3)
- 2/(x^2 − 9) * (x − 3)(x + 3) = 2
- 4/(x + 3) * (x − 3)(x + 3) = 4(x − 3)

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

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

5) Check x = −14 in the original equation
Compute each side for x = −14:
- x − 3 = −17, x + 3 = −11, x^2 − 9 = 187.
Left side: 6/(−17) − 2/187 = −6/17 − 2/187 = −66/187 − 2/187 = −68/187.
Right side: 4/(−11) = −4/11 = −68/187.
They match, and x = −14 is not excluded (not ±3).

Final solution: x = −14. (Excluded values: x = 3, x = −3.)