Question

solve \\(\frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x}\\) by rewriting the equation as a proportion. which proportion is equivalent to the original equation? \\(\frac{x - 1}{2x} = \frac{x^2 - 7x + 10}{4x}\\) \\(\frac{x + 1}{2x} = \frac{x^2 - 7x + 10}{4x}\\) \\(\frac{x + 2}{2x} = \frac{x^2 - 7x + 10}{4x}\\)

Explanation:

Step1: Combine left - hand side terms

We have the left - hand side of the equation \(\frac{1}{2}+\frac{1}{2x}\). To add these two fractions, we need a common denominator. The common denominator of 2 and \(2x\) is \(2x\).
Rewrite \(\frac{1}{2}\) with the denominator \(2x\): \(\frac{1}{2}=\frac{x}{2x}\) (multiplying the numerator and denominator by \(x\)).
Then \(\frac{1}{2}+\frac{1}{2x}=\frac{x}{2x}+\frac{1}{2x}\).
According to the rule of adding fractions with the same denominator \(\frac{a}{c}+\frac{b}{c}=\frac{a + b}{c}\), we get \(\frac{x + 1}{2x}\).

Step2: Compare with the right - hand side

The original equation is \(\frac{1}{2}+\frac{1}{2x}=\frac{x^{2}-7x + 10}{4x}\), and after combining the left - hand side, we have \(\frac{x + 1}{2x}=\frac{x^{2}-7x + 10}{4x}\).

Answer:

\(\frac{x + 1}{2x}=\frac{x^{2}-7x + 10}{4x}\) (the second option)