Solve the equation. \( \frac{x+6}{6}=2-\frac{x+5}{5} \) What is the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \). (Type an integer or a fraction.) B. The solution set is \( \{x \mid x \) is a real number \( \} \). C. The solution set is \( \varnothing \).

To solve the equation \( \frac{x+6}{6}=2-\frac{x+5}{5} \), we first eliminate the fractions by multiplying both sides by 30 (the least common multiple of 6 and 5): \( 30 \cdot \frac{x+6}{6} = 30 \cdot \left(2 - \frac{x+5}{5}\right) \) This simplifies to: \( 5(x + 6) = 60 - 6(x + 5) \) Expanding both sides gives: \( 5x + 30 = 60 - 6x - 30 \) Combining like terms on the right side leads to: \( 5x + 30 = 30 - 6x \) Now, adding \( 6x \) to both sides results in: \( 5x + 6x + 30 = 30 \) Thus: \( 11x + 30 = 30 \) Subtracting 30 from both sides yields: \( 11x = 0 \) Finally, dividing both sides by 11 gives: \( x = 0 \) So the solution set is \( \{0\} \). Now, let's explore more on this topic: In the world of algebra, equations like this are foundational building blocks! Solving them often requires a good grasp of fraction manipulation, variable isolation, and basic arithmetic. Each solution you find is like unraveling a little mystery, revealing the value that satisfies the equation. For those new to solving equations, a common mistake is forgetting to apply operations to both sides of the equation equally. Always check your steps and simplify carefully! It’s great practice to verify your solution by plugging it back into the original equation to ensure both sides are equal. It’s like double-checking your homework before handing it in!