Question

1. \\(\frac{1}{6x} - \frac{2x - 12}{3x} = \frac{1}{6}\\)

Explanation:

Step1: Find common denominator

The denominators are \(6x\), \(3x\), and \(6\). The least common denominator (LCD) is \(6x\). Multiply each term by \(6x\) to eliminate denominators:
\(6x\times\frac{1}{6x}-6x\times\frac{2x - 12}{3x}=6x\times\frac{1}{6}\)
Simplify each term:
\(1 - 2(2x - 12)=x\)

Step2: Expand and simplify

Expand the left - hand side:
\(1-4x + 24=x\)
Combine like terms:
\(25-4x=x\)

Step3: Solve for x

Add \(4x\) to both sides:
\(25=x + 4x\)
\(25 = 5x\)
Divide both sides by \(5\):
\(x=\frac{25}{5}=5\)

Step4: Check the solution

Substitute \(x = 5\) into the original equation:
Left - hand side: \(\frac{1}{6\times5}-\frac{2\times5 - 12}{3\times5}=\frac{1}{30}-\frac{10 - 12}{15}=\frac{1}{30}+\frac{2}{15}=\frac{1 + 4}{30}=\frac{5}{30}=\frac{1}{6}\)
Right - hand side: \(\frac{1}{6}\)
Since the left - hand side equals the right - hand side, \(x = 5\) is a valid solution.

Answer:

\(x = 5\)