Divide both sides: $ (1.125)^{n-1} > 95 / 64 = 1.484375 $

["Title: How to Solve $ (1.125)^{n-1} > \frac{95}{64} $: A Step-by-Step Guide to Dividing Both Sides", "Meta Description:\nLearn how to solve the inequality $ (1.125)^{n-1} > \frac{95}{64} $ by dividing both sides. Discover step-by-step methods, real-world applications, and key mathematical insights for accurate, clean solutions.", "---", "### Understanding the Inequality: $ (1.125)^{n-1} > \frac{95}{64} $", "When solving exponential inequalities like $ (1.125)^{n-1} > \frac{95}{64} $, direct comparison can be tricky. To simplify, we use a key algebraic principle: dividing both sides by a positive number preserves inequality direction — a foundational step in isolating the variable.", "First, rewrite $ \frac{95}{64} $ as a decimal approximation to understand the inequality clearly:\n$$\n\frac{95}{64} = 1.484375\n$$\nSo the inequality becomes:\n$$\n(1.125)^{n-1} > 1.484375\n$$", "### Why Divide Both Sides?", "The base $ 1.125 $ is positive and greater than 1. Since exponential functions with bases $ > 1 $ are strictly increasing, the exponent $ n-1 $ must increase to make the left-hand side exceed $ 1.484375 $. To isolate $ n-1 $, log-based techniques usually apply, but dividing both sides by $ 1.125 $ simplifies the coefficient and sets the stage for logarithmic or progression-based methods.", "---", "### Step-by-Step Solution Dividing Both Sides", "1. Start with the original inequality:\n$$\n(1.125)^{n-1} > 1.484375\n$$", "2. Divide both sides by $ 1.125 $:\nSince $ 1.125 > 0 $, division maintains inequality direction:\n$$\n(1.125)^{n-1} \div 1.125 > \frac{1.484375}{1.125}\n$$", "3. Simplify the left-hand side:\n$$\n(1.125)^{n-2} > \frac{1.484375}{1.125}\n$$", "4. Calculate the right-hand side:\n$$\n\frac{1.484375}{1.125} = 1.325\n$$", "5. Now solve for $ n-2 $ using logarithms:\nTake the logarithm of both sides (base 10 or natural log work, but use base 10 here for simplicity):\n$$\n\log\left((1.125)^{n-2}\right) > \log(1.325)\n$$", "$$\n(n - 2) \cdot \log(1.125) > \log(1.325)\n$$", "6. Isolate $ n - 2 $:\n$$\nn - 2 > \frac{\log(1.325)}{\log(1.125)}\n$$", "7. Compute values:\nUsing logarithms ( calculator or log table):\n- $ \log(1.325) \approx 0.1223 $\n- $ \log(1.125) \approx 0.0512 $", "So:\n$$\nn - 2 > \frac{0.1223}{0.0512} \approx 2.386\n$$", "8. Solve for $ n $:\n$$\nn > 2 + 2.386 = 4.386\n$$", "---", "### Final Answer", "Since $ n $ must exceed $ 4.386 $, and assuming $ n $ is a real number (or integer), the smallest integer solution is:\n$$\nn \geq 5\n$$\nBut the precise solution set is:\n$$\nn > 4.386\n$$\nExpressed in boxed form:\n$$\n\boxed{n > 4.386}\n$$\nor as a decimal inequality:\n$$\nn > 4.386\n$$", "---", "### Why This Dividing Method Matters", "Dividing both sides preserves inequality orientation — a critical algebraic rule — enabling clearer progression when combining logs or analyzing growth. While logarithms ultimately solve the exponent, dividing first streamlines expression, making advanced techniques more intuitive.", "---", "### Practical Applications", "This type of inequality appears in:\n- Exponential growth models, e.g., bacterial growth, compound interest, or population studies\n- Engineering and physics, where thresholds must not be crossed under increasing performance\n- Decision thresholds in economics or risk management, where a value must exceed a critical boundary", "---", "### Summary", "- Start with $ (1.125)^{n-1} > \frac{95}{64} $\n- Divide both sides by $ 1.125 $ to simplify\n- Apply logarithms to isolate the exponent\n- Solve numerically for precise bounds\n- Interpret the result within real-world contexts", "Mastering “divide both sides” instills clarity and precision — key assets in solving exponential and logarithmic challenges.", "---", "Keywords:\n$ (1.125)^{n-1} > \frac{95}{64} $, divide both sides inequality, exponential inequality solution, logarithmic steps, real exponent solutions, mathematical method, growth model inequality, divide side inequality, 1.125 exponential inequality", "---", "Further Reading:\n- Logarithmic methods for exponential inequalities\n- Analyzing discrete vs continuous growth thresholds\n- Step-by-step guide to solving exponential equations with base > 1", "---", "Stay algebraically sharp — understanding division and exponents empowers smarter problem-solving in science, finance, and engineering."]