Question

1. \\(\frac{6x - 6}{x^2} = \frac{1}{2x} + \frac{x + 2}{2x^2}\\)

Explanation:

Step1: Find common denominator

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

Step2: Expand and simplify left side

Expand \(2(6x - 6)\): \(12x-12=x+(x + 2)\)

Step3: Simplify right side

Combine like terms on the right: \(12x-12 = 2x+2\)

Step4: Solve for x

Subtract \(2x\) from both sides: \(10x-12 = 2\)
Add 12 to both sides: \(10x=14\)
Divide by 10: \(x=\frac{14}{10}=\frac{7}{5}\)

Step5: Check for restrictions

Original denominators: \(x^2
eq0\) (so \(x
eq0\)) and \(2x
eq0\) (so \(x
eq0\)). \(\frac{7}{5}
eq0\), so it's valid.

Answer:

\(x = \frac{7}{5}\)