How many solutions does the following equation have? \( -9(x+6)=-9 x+108 \) Choose 1 answer: (A) No solutions (B) Exactly one solution (C) Infinitely many solutions

To solve the equation \( -9(x+6)=-9 x+108 \), let's simplify it. Expanding the left side gives us: \(-9x - 54 = -9x + 108\). Next, if we add \(9x\) to both sides, we get: \(-54 = 108\). This statement is false, meaning there are no values of \(x\) that can satisfy the equation. Therefore, the answer is \( \text{(A) No solutions} \). Now let's delve into real-world applications! This type of equation may arise when analyzing financial transactions, such as balancing income and expenditures. If you’re budgeting and want to set up a formula to track your spending versus earnings, understanding how to simplify and solve these types of equations is crucial for achieving your financial goals! Another fun aspect is how equations like this can inspire creativity! When students encounter contradictions like \( -54 = 108\), it’s an opportunity to explore the concept of logical fallacies in mathematics, leading to questions about how we can sometimes expect certain outcomes to be true when they simply aren’t. Embrace those “lightbulb” moments!