Step-by-Step Solution

Solve the equation: $\frac{1}{3x} + \frac{4x}{6} = 1 \quad (x \ne 0)$

Step-by-Step Solution

Step 1: Simplify and clear fractions

Multiply the entire equation by the common denominator $6x$: $6x \left(\frac{1}{3x}\right) + 6x \left(\frac{4x}{6}\right) = 6x(1)$

$2 + 4x^2 = 6x$

Step 2: Write in standard quadratic form

$4x^2 - 6x + 2 = 0$

Divide by $2$: $2x^2 - 3x + 1 = 0$

Step 3: Factorize

$2x^2 - 2x - x + 1 = 0$ $2x(x - 1) - 1(x - 1) = 0$ $(2x - 1)(x - 1) = 0$

$2x - 1 = 0 \implies x = \frac{1}{2}$ $x - 1 = 0 \implies x = 1$

Step 4: Check solutions

• For $x = \frac{1}{2}$: $\dfrac{1}{3(1/2)} + \dfrac{4(1/2)}{6} = \dfrac{2}{3} + \dfrac{1}{3} = 1$ $\checkmark$
• For $x = 1$: $\dfrac{1}{3(1)} + \dfrac{4(1)}{6} = \dfrac{1}{3} + \dfrac{2}{3} = 1$ $\checkmark$

$\boxed{\text{Solution Set: } \left\{\frac{1}{2}, \; 1\right\}}$