To solve the equation \( \frac{5}{x} + 8 = \frac{3}{x} \), we start by eliminating the fractions. First, we can multiply both sides of the equation by \( x \) (assuming \( x \neq 0 \)) to simplify our work:
Step 1: Multiply through by \( x \)
x(\frac{5}{x} + 8) = x(\frac{3}{x}) This simplifies to:
5 + 8x = 3 Step 2: Rearrange the equation
Next, we can rearrange this equation to isolate terms involving \( x \):
8x = 3 - 5 Which simplifies to:
8x = -2 Step 3: Solve for \( x \)
Now we can divide both sides by 8:
x = \frac{-2}{8} = \frac{-1}{4} Thus, the solution is \( x = -\frac{1}{4} \).
Step 4: Check for extraneous solutions
We need to check if this solution is extraneous. We substitute \( x = -\frac{1}{4} \) back into the original equation:
\frac{5}{-\frac{1}{4}} + 8 = \frac{3}{-\frac{1}{4}} This can be simplified as follows:
-20 + 8 = -12 This simplifies to:
-12 = -12 Since both sides are equal, the solution\( x = -\frac{1}{4} \) is indeed valid and not extraneous.
In conclusion: The solution to the equation \( \frac{5}{x} + 8 = \frac{3}{x} \) is \( x = -\frac{1}{4} \), and it is not an extraneous solution.