Question

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)

1/x + 1/x-8 = 1/3

Answer 1

Step 1 — Undefined values:
Denominators are x, x − 8, and 3, so x ≠ 0 and x ≠ 8.

Step 2 — Factor denominators and state LCD:
Denominators: x, (x − 8), 3. LCD = 3x(x − 8).

Step 3 — Multiply each term by the LCD:
Multiply both sides of 1/x + 1/(x − 8) = 1/3 by 3x(x − 8):
3x(x − 8)*(1/x) + 3x(x − 8)*(1/(x − 8)) = 3x(x − 8)*(1/3).

Step 4 — Simplify (no denominators):
3(x − 8) + 3x = x(x − 8).

Expand: 3x − 24 + 3x = x^2 − 8x
So: 6x − 24 = x^2 − 8x
Bring all terms to one side: 0 = x^2 − 14x + 24.

Step 5 — Solve for x:
Factor: x^2 − 14x + 24 = (x − 2)(x − 12) = 0
Thus x = 2 or x = 12.

Step 6 — Check in original (watch for extraneous solutions):
x = 2: 1/2 + 1/(2 − 8) = 1/2 + 1/(−6) = 1/2 − 1/6 = 1/3 ✓
x = 12: 1/12 + 1/(12 − 8) = 1/12 + 1/4 = 1/12 + 3/12 = 1/3 ✓

Neither solution is excluded by x ≠ 0,8, so no extraneous solutions.

Step 7 — Final solution(s):
x = 2 or x = 12.