# How do you solve #10/(x + 4) = 15/(4x + 4)# and find any extraneous solutions?

Solution:

issue, so a superfluous solution is not needed.

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To solve the equation 10/(x + 4) = 15/(4x + 4) and find any extraneous solutions, we can start by cross-multiplying to eliminate the fractions. This gives us 10(4x + 4) = 15(x + 4). Simplifying this equation, we get 40x + 40 = 15x + 60. By combining like terms, we have 40x - 15x = 60 - 40, which simplifies to 25x = 20. Dividing both sides by 25, we find x = 20/25, which reduces to x = 4/5. To check for extraneous solutions, we substitute x = 4/5 back into the original equation. We have 10/(4/5 + 4) = 15/(4(4/5) + 4). Simplifying this equation, we get 10/(24/5) = 15/(16/5). By multiplying both sides by the reciprocal of the denominator, we have 10 * (5/24) = 15 * (5/16). This simplifies to 50/24 = 75/16. However, these fractions are not equal, indicating that x = 4/5 is an extraneous solution. Therefore, there are no valid solutions to the equation.

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