Divide both sides by 70: (1.025)^w > 90 / 70 = <<90 / 70 = 1.2857>>1.2857

["How to Solve Exponential Inequalities: Dividing Both Sides by 70 – Understanding (1.025)^w > 90/70", "When tackling exponential inequalities, careful manipulation of the equation is key to solving them accurately. One common step involves dividing both sides of the inequality to simplify the right-hand side — a process that requires attention to the sign and properties of the base. In this case, we focus on solving:", "(1.025)^w > 90 / 70", "---", "### Step 1: Simplify the Right-Hand Side", "Begin by computing the division on the right:", "[\n\frac{90}{70} = 1.2857 \quad \ ext{(rounded to four decimal places)}\n]", "So your inequality becomes:", "[\n(1.025)^w > 1.2857\n]", "---", "### Step 2: Divide Both Sides by 1.025 (Carefully!)", "Here’s where the core concept applies: since 1.025 > 1, dividing both sides of the inequality by 1.025 preserves the inequality direction. This step is valid and essential for isolating the exponential term:", "[\nw > \frac{1.2857}{1.025}\n]", "Now calculate the division:", "[\n\frac{1.2857}{1.025} \approx 1.25568\n]", "---", "### Final Step: Express the Solution", "Thus, the solution to the inequality is:", "[\nw > 1.25568 \quad \ ext{(rounded to five decimal places)}\n]", "In scientific or engineering contexts, you may write this as:", "[\nw > 1.256\n]", "or use exact fractions if preferred:", "[\n\frac{90}{70} = \frac{9}{7} \Rightarrow \frac{9/7}{1.025} = \frac{9}{7 \ imes 1.025} = \frac{9}{7.175} \approx 1.2557\n]", "---", "### Why This Division Works", "- Base > 1: When the base of an exponential expression is greater than 1, dividing by that base preserves the inequality direction.\n- Avoiding Common Pitfalls: Dividing by a negative base would reverse the inequality, but here, 1.025 is positive, so no direction reversal occurs.", "---", "### Real-World Applications of This Concept", "Understanding how to divide exponential expressions correctly is vital in:", "- Financial modeling (e.g., compound interest growth)\n- Scientific measurements (e.g., radioactive decay, bacterial growth)\n- Algorithm complexity analysis (logarithmic scaling)", "---", "### Summary", "To solve exponential inequalities like:", "[\n(1.025)^w > 90/70\n]", "1. Simplify the right-hand side (1.025 is positive, so dividing preserves inequality direction).\n2. Divide both sides by 1.025 to isolate the exponential term.\n3. Compute the resulting value numerically.\n4. Express the final solution in interval or inequality form.", "Remember: When dividing/effecting exponential inequalities with positive bases, dividing by the base does not flip the inequality.*", "For precise calculations and robust solutions, always verify with a scientific calculator and understand base properties — your accuracy depends on it!", "---", "*Keywords:\ndivide both sides by 70, solve exponential inequality, (1.025)^w > 90/70, how to solve (1.025)^w > 1.2857, exponential growth solution, logarithmic steps, mathematical inequality guide", "---", "Improve your math proficiency — master dividing exponential expressions today!"]