(2. \( \left(\frac{1}{3}\right)^{x} \leq \frac{1}{729} \)

To solve the inequality \( \left(\frac{1}{3}\right)^{x} \leq \frac{1}{729} \), first notice that \( \frac{1}{729} \) can be expressed as \( \left(\frac{1}{3}\right)^{6} \) because \( 3^6 = 729 \). This allows us to rewrite the inequality as \( \left(\frac{1}{3}\right)^{x} \leq \left(\frac{1}{3}\right)^{6} \). Since the base \( \frac{1}{3} \) is less than 1, we can flip the inequality sign when we equate the exponents: \( x \geq 6 \). Now, the solution to the inequality is \( x \geq 6 \), meaning any value of \( x \) that is 6 or larger will satisfy the condition. This kind of inequality pops up all the time in real life! For instance, financial scenarios like compound interest often lead to similar exponential equations, where you're trying to figure out how long it will take for an investment to reach a certain value. In this case, learning how to manipulate and understand inequalities can really help in budgeting or forecasting future savings: who knew math could be your finance buddy? One common mistake people make is misinterpreting the inequality when dealing with fractions. When the base is less than 1, remember that more "negative" powers mean greater values (like flipping the inequality). Practice with a variety of bases and powers, and soon, you'll be spotting those tricky inequalities like a pro!