Questions: Add. 4/(x-2) + 3/(5x-10) Simplify your answer as much as possible.

Transcript text: Add. \[ \frac{4}{x-2}+\frac{3}{5 x-10} \] Simplify your answer as much as possible.

Solution

Step 1: Identify a Common Denominator

The given expression is:

\[ \frac{4}{x-2} + \frac{3}{5x-10} \]

First, notice that the denominator of the second fraction, \(5x - 10\), can be factored:

\[ 5x - 10 = 5(x - 2) \]

Thus, the expression becomes:

\[ \frac{4}{x-2} + \frac{3}{5(x-2)} \]

The common denominator for these fractions is \(5(x-2)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite the first fraction with the common denominator:

\[ \frac{4}{x-2} = \frac{4 \cdot 5}{(x-2) \cdot 5} = \frac{20}{5(x-2)} \]

The second fraction is already in terms of the common denominator:

\[ \frac{3}{5(x-2)} \]

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them:

\[ \frac{20}{5(x-2)} + \frac{3}{5(x-2)} = \frac{20 + 3}{5(x-2)} = \frac{23}{5(x-2)} \]

The simplified expression is:

\[ \boxed{\frac{23}{5(x-2)}} \]

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