Question

Simplify the expression. $$\frac{x-2}{2 x-10}+\frac{x+3}{x-5}$$

Short answer

\(\frac{3x}{2} - 1\)

Step-by-step solution

Identify the common denominator

The denominator of the first term is \(2x-10\), which can be written as \(2*(x-5)\). Now, it's clear to see the common denominator of both fractions, which is \(x-5\).

Rewrite the fractions with the common denominator

Now, rewrite the first fraction as follows: \(\frac{(x-2)/2}{(2x-10)/2} = \frac{x-2}{2*(x-5)}\). Now, both fractions are \(\frac{x-2}{2*(x-5)}\) and \(\frac{x+3}{x-5}\).

Simplify the first fraction

Now, simplify the first fraction to get: \(\frac{x-2}{2*(x-5)} = \frac{x-2}{2} * \frac{1}{x-5}\). This fractions simplify to: \(\frac{x-2}{2} * \frac{1}{x-5} = \frac{x-2}{2} * \frac{1}{x-5} = \frac{x}{2} - \frac{2}{2} = \frac{x}{2} - 1\).

Add the two fractions

Now we can add the two simplified fractions because they have the common denominator: \(\frac{x}{2} - 1 + \frac{x+3}{x-5} = \frac{3x}{2} - 1\).