Solve: 2.5 × (0.88)^u < 1.5
["# How to Solve: 2.5 × (0.88)^u < 1.5 – A Step-by-Step Explanation", "Understanding how to solve exponential inequality is fundamental in algebra and helps build a strong foundation for more advanced math topics. In this article, we’ll walk through solving the inequality:", "2.5 × (0.88)^u < 1.5", "We’ll break down each step clearly, making it easy to follow and apply. 📈", "---", "## Step 1: Isolate the Exponential Term", "Start by isolating the exponential component. Divide both sides of the inequality by 2.5:", "$$\n(0.88)^u < \frac{1.5}{2.5} = 0.6\n$$", "Now your inequality looks like:", "$$\n(0.88)^u < 0.6\n$$", "---", "## Step 2: Take the Logarithm of Both Sides", "To solve for the exponent ( u ), apply logarithms. Since the base ( 0.88 ) is between 0 and 1, logarithms will work well. Use natural log (ln) or base 10 log — both are valid. We’ll use base 10 for simplicity:", "$$\n\log\left((0.88)^u\right) < \log(0.6)\n$$", "Use the logarithmic identity (\log(a^b) = b\log(a)):", "$$\nu \cdot \log(0.88) < \log(0.6)\n$$", "---", "## Step 3: Solve for ( u )", "Since ( \log(0.88) ) is negative (because ( 0.88 < 1 )), dividing both sides reverses the inequality:", "$$\nu > \frac{\log(0.6)}{\log(0.88)}\n$$", "Now compute the values:", "- ( \log(0.6) \approx -0.2218 )\n- ( \log(0.88) \approx -0.0555 )", "Substitute:", "$$\nu > \frac{-0.2218}{-0.0555} \approx 3.997\n$$", "---", "## Step 4: Interpret the Result", "The solution is:", "$$\nu > 3.997\n$$", "This means any real number ( u ) greater than about 3.997 satisfies the original inequality.", "---", "## Summary", "To solve:", "2.5 × (0.88)^u < 1.5,", "1. Divide both sides by 2.5 → ( (0.88)^u < 0.6 )\n2. Take log of both sides → ( u \cdot \log(0.88) < \log(0.6) )\n3. Divide by negative log(0.88) → flips inequality → ( u > \frac{\log(0.6)}{\log(0.88)} )\n4. Calculate → ( u > 3.997 )", "---", "## Why This Matters", "Exponential inequalities appear in many fields — from finance (decay models) to science (radioactive decay, population dynamics). Knowing how to solve them empowers you to model real-world phenomena accurately.", "---", "Keywords: solve exponential inequality, solve 2.5 × (0.88)^u < 1.5, logarithmic inequality, exponential functions, math tutorial, u > 3.997, mathematical steps explanation", "---", "If you want to visualize this inequality, plotting ( y = 2.5 \ imes (0.88)^u ) and ( y = 1.5 ) helps confirm where the exponential curve lies below 1.5. Use graphing tools or calculators to see this in action!"]
