Question

simplifying each side of the equation results in $x^2 - 3x - 4 = x^2 - 5x + 6$. find the solution: $\frac{x + 2}{3x} - \frac{1}{x - 2} = \frac{x - 3}{3x}$

Explanation:

Step1: Eliminate common denominator

Multiply all terms by $3x(x-2)$ to cancel denominators:
$$(x+2)(x-2) - 3x = (x-3)(x-2)$$

Step2: Expand all binomials

Use FOIL method to expand each product:
$$x^2 - 4 - 3x = x^2 - 5x + 6$$

Step3: Cancel $x^2$ from both sides

Subtract $x^2$ from each side:
$$-3x - 4 = -5x + 6$$

Step4: Isolate $x$ terms

Add $5x$ to both sides:
$$2x - 4 = 6$$

Step5: Isolate constant terms

Add 4 to both sides:
$$2x = 10$$

Step6: Solve for $x$

Divide both sides by 2:
$$x = 5$$

Step7: Verify solution

Substitute $x=5$ into original equation to confirm no division by zero and equality holds:
$$\frac{5+2}{3(5)} - \frac{1}{5-2} = \frac{7}{15} - \frac{5}{15} = \frac{2}{15}, \quad \frac{5-3}{3(5)} = \frac{2}{15}$$

Answer:

$x=5$