Question

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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).

Answer 1

You already have the cleared-denominator equation. I'll show every step from the original rational equation to the solution.

Original equation:
[6(x + 3) − 2] / [(x − 3)(x + 3)] = [4(x − 3)] / [(x − 3)(x + 3)]

Step 1 — Domain restriction:
Denominators (x − 3)(x + 3) ≠ 0, so x ≠ 3 and x ≠ −3.

Step 2 — Clear denominators:
Multiply both sides by (x − 3)(x + 3):
6(x + 3) − 2 = 4(x − 3)

Step 3 — Expand both sides:
Left: 6(x + 3) − 2 = 6x + 18 − 2 = 6x + 16
Right: 4(x − 3) = 4x − 12

So 6x + 16 = 4x − 12.

Step 4 — Solve for x:
Subtract 4x from both sides: 2x + 16 = −12
Subtract 16 from both sides: 2x = −28
Divide by 2: x = −14

Step 5 — Check against domain and verify:
x = −14 is allowed (not 3 or −3). Substitute to check:
Numerators: 6(x+3)−2 = 6(−11)−2 = −66−2 = −68; 4(x−3) = 4(−17) = −68.
Denominator (same both sides) = (−14−3)(−14+3) = (−17)(−11) = 187.
Both sides = −68/187, so the solution is valid.

Final answer: x = −14 (with x ≠ 3, −3 as excluded values).