To solve the equation 5/x + 8 = 3/x, we first eliminate the fractions by multiplying both sides of the equation by x (assuming x ≠ 0):
x(5/x + 8) = x(3/x)
This simplifies to:
5 + 8x = 3
Next, we isolate the variable x by moving 5 to the right side:
8x = 3 – 5
8x = -2
Now, we divide both sides by 8 to solve for x:
x = -2/8
x = -1/4
Now, let’s check if this solution is extraneous by substituting -1/4 back into the original equation.
Substituting into 5/x + 8 = 3/x:
5/(-1/4) + 8 = 3/(-1/4)
Calculating the left side:
5 ÷ (-1/4) = 5 * (-4/1) = -20
So, the left side becomes:
-20 + 8 = -12
Now, calculating the right side:
3 ÷ (-1/4) = 3 * (-4/1) = -12
The right side is also:
-12
Since both sides of the equation are equal (-12 = -12), the solution x = -1/4 is valid and not extraneous.
In conclusion, the solution to the equation is x = -1/4 and it is not an extraneous solution.