\frac{100}{n^2 + 1} < 5
["Understanding the Inequality: When Does $\frac{100}{n^2 + 1} < 5 Hold True?", "When faced with the inequality\n$$\n\frac{100}{n^2 + 1} < 5,\n$$\nsolving it correctly is essential, especially for students, educators, and math enthusiasts. In this article, we’ll explore how to solve this inequality step-by-step, explain its meaning, and provide practical insights into when the inequality is true.", "---", "### What Does the Inequality Mean?", "We are comparing the rational expression\n$$\n\frac{100}{n^2 + 1}\n$$\nto the constant 5. This inequality tells us for which values of $ n $ the expression on the left becomes smaller than 5.", "Note: Since $ n^2 + 1 $ is always positive (even when $ n $ is negative), the denominator is always greater than 1, ensuring the fraction is always positive and defined for all real $ n $. This means we are working in the domain of all real numbers.", "---", "### Step-by-Step Solution", "Start with:\n$$\n\frac{100}{n^2 + 1} < 5\n$$", "Multiply both sides by $ n^2 + 1 $ (which is positive, so the inequality sign stays the same):\n$$\n100 < 5(n^2 + 1)\n$$", "Expand the right-hand side:\n$$\n100 < 5n^2 + 5\n$$", "Subtract 5 from both sides:\n$$\n95 < 5n^2\n$$", "Divide both sides by 5:\n$$\n19 < n^2\n$$", "Rewriting:\n$$\nn^2 > 19\n$$", "Take square roots of both sides (remembering both positive and negative roots):\n$$\n|n| > \sqrt{19}\n$$", "Since $ \sqrt{19} \approx 4.36 $, the solution is:\n$$\nn < -\sqrt{19} \quad \ ext{or} \quad n > \sqrt{19}\n$$", "---", "### The Final Answer", "The inequality\n$$\n\frac{100}{n^2 + 1} < 5\n$$\nholds for all real numbers $ n $ satisfying\n$$\nn < -\sqrt{19} \quad \ ext{or} \quad n > \sqrt{19}.\n$$", "Numerically approximated:\n$$\nn < -4.36 \quad \ ext{or} \quad n > 4.36\n$$", "---", "### Visualizing the Solution", "Graphing $ y = \frac{100}{n^2 + 1} $ and $ y = 5 $, you’ll find the function lies below 5 when $ |n| > \sqrt{19} $, confirming our algebraic result.", "---", "### Why This Matters", "This inequality exemplifies how rational expressions behave asymptotically—approaching zero as $ n $ grows large—and how inequalities involving squares reflect symmetry about zero. Understanding it helps in math coursework, data modeling, and interpreting real-world data curves.", "---", "### FAQ: Common Questions", "Q: Can $ n $ be any real number?\nA: Yes, since $ n^2 + 1 > 0 $ always, the expression is defined everywhere, and the inequality solution applies to all real $ n $.", "Q: What values of $ n $ make the expression equal to 5?\nA: When $ n^2 = 19 $, $ \frac{100}{19 + 1} = \frac{100}{20} = 5 $. So equality occurs at $ n = \pm\sqrt{19} $.", "Q: How does this relate to the function’s maximum?\nA: The expression $ \frac{100}{n^2 + 1} $ peaks at $ n = 0 $, where it equals 100. As $ |n| $ increases, the fraction decreases—crossing 5 at $ n = \pm\sqrt{19} $.", "---", "Summary:\nTo solve $ \frac{100}{n^2 + 1} < 5 $, determine when $ n^2 > 19 $. The inequality holds true for all $ n $ such that $ |n| > \sqrt{19} $. This understanding strengthens your grasp of rational inequalities and functions—key tools in algebra and calculus.", "---", "Try solving similar inequalities yourself, and explore how limits and asymptotes influence inequality solutions!"]
