Question

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

Step 1: Find the undefined terms of the equation.Step 2: Factor all denominators and state the LCD.Step 3: Correctly rewrite the equation using the LDC.Step 4: Simplify the equation and show your final equation that has no denominator.Step 5: Solve for xStep 6: Plug in your value(s) for x in the original equation to check for extraneous solution(s).Step 7: State your final solution(s).(8 points)

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

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

Step 1 — Undefined terms
Denominators are x − 3, x^2 − 9, and x + 3. So x ≠ 3 and x ≠ −3.

Step 2 — Factor denominators and give the LCD
x^2 − 9 = (x − 3)(x + 3).
LCD = (x − 3)(x + 3).

Step 3 — Rewrite each term over the LCD
6/(x − 3) = 6(x + 3)/[(x − 3)(x + 3)],
−2/(x^2 − 9) = −2/[(x − 3)(x + 3)],
4/(x + 3) = 4(x − 3)/[(x − 3)(x + 3)].

So the equation becomes
[6(x + 3) − 2]/[(x − 3)(x + 3)] = [4(x − 3)]/[(x − 3)(x + 3)].

Step 4 — Clear denominators and simplify (final equation with no denominator)
Multiply both sides by (x − 3)(x + 3):
6(x + 3) − 2 = 4(x − 3).

Expand and simplify:
6x + 18 − 2 = 4x − 12
6x + 16 = 4x − 12

Step 5 — Solve for x
6x − 4x = −12 − 16
2x = −28
x = −14

Step 6 — Check in the original equation
Left side:
6/(−14 − 3) − 2/( (−14)^2 − 9 ) = 6/(−17) − 2/(196 − 9)
= −6/17 − 2/187
Convert −6/17 to seventeenths of 187: −6/17 = −66/187, so LHS = −66/187 − 2/187 = −68/187.

Right side:
4/(−14 + 3) = 4/(−11) = −4/11 = −68/187.

LHS = RHS, and x = −14 is not forbidden by the domain.

Step 7 — Final solution
x = −14 (no extraneous solutions).