Case 3: \( y < -1 \), numerator and denominator both negative → positive.

["SEO Article: Understanding Case 3—When Both Numerator and Denominator Are Negative, Resulting in a Positive Fraction", "---", "### Introduction", "In algebra, fractions are one of the most fundamental expressions students encounter. Among the various cases that determine the sign of a rational expression, Case 3 is both intuitive and essential for mastering sign analysis: when both the numerator and denominator are negative, the fraction is positive. This article delves into Case 3, explaining why the expression becomes positive and how this principle plays a crucial role in algebra, calculus, and real-world applications.", "---", "### What Is Case 3 of Sign Analysis?", "In rational expressions, expressions like ( \frac{f(x)}{g(x)} ) can take positive or negative values depending on the signs of the numerator ( f(x) ) and the denominator ( g(x) ). There are four major cases to analyze:", "1. Both numerator and denominator positive (positive/positive = positive)\n2. Numerator negative, denominator positive (negative/positive = negative)\n3. Both numerator and denominator negative (negative/negative = positive)\n4. Numerator positive, denominator negative (positive/negative = negative)", "Case 3 specifically occurs when:", "[\n\ ext{Numerator} < 0 \quad \ ext{and} \quad \ ext{Denominator} < 0 \quad \Rightarrow \quad \frac{\ ext{Numerator}}{\ ext{Denominator}} > 0\n]", "This boxed result — a positive value — is a critical concept in solving inequalities, graphing rational functions, and understanding function behavior.", "---", "### Why is Case 3 Positive?", "If both the numerator and the denominator are negative, dividing a negative by a negative cancels out the negative signs:", "[\n\frac{-a}{-b} = \frac{a}{b}, \quad \ ext{where } a > 0 \ ext{ and } b > 0\n]", "Since both parts of the fraction are positive, the overall value is positive. This is consistent with the rules of arithmetic and extends naturally into algebra and beyond.", "Example:\nEvaluate ( \frac{-5}{-3} )", "[\n\frac{-5}{-3} = \frac{5}{3} = 1.\overline{6}\n]", "Clearly, the result is positive despite both inputs being negative — demonstrating Case 3 clearly.", "---", "### Practical Implications and Applications", "Understanding Case 3 is vital in multiple areas:", "- Solving rational inequalities: Identifying intervals where the expression is positive helps determine solution sets for inequalities like ( \frac{x+2}{x-4} < 0 ).", "- Graphing rational functions: Knowing sign patterns—especially where the function is positive—directs how graphs behave across asymptotes and intercepts.", "- Physics and Engineering: In modeling real-world phenomena such as force ratios, electrical resistance, or rates, negative and positive signs represent direction or magnitude, and Case 3 ensures accurate interpretation.", "---", "### How to Identify Case 3 Easily", "To spot Case 3 in expressions:", "1. Factor the numerator and denominator completely.\n2. Write the fraction in simplified form, conserving signs.\n3. Check that both parts are negative. (Not just one being negative!)", "Always remember:", "> When signs match, the result is positive — even in negative divisions.", "---", "### Summary", "Case 3 — numerator < 0 and denominator < 0 — results in a positive fraction. This principle is one of the cornerstones of rational expression analysis. Mastering it enables students and learners to confidently interpret signs, solve complex inequalities, and apply algebraic reasoning in scientific and engineering contexts.", "Remember:\nNegative ÷ Negative = Positive", "So next time you encounter a negative-number fraction with both parts negative, smile — because the answer is indeed positive.", "---", "### Keywords for SEO Optimization", "- Case 3 sign meaning\n- Rational expressions positive fraction\n- How both numerator and denominator negative yields positive\n- Algebra fraction sign rules\n- Solve rational inequalities with sign analysis\n- Negative over negative fraction equals positive", "---", "By understanding Case 3, you unlock deeper insight into rational expressions and build a stronger foundation for advanced mathematics. Don’t overlook this simple but powerful rule — it’s essential for academic success and real-world problem solving!", "---", "Keywords: case 3 sign of fraction, numerator and denominator both negative, positive rational expression, algebra sign rules, rational function analysis"]