For \( x < -3 \), say \( x = -4 \): \( rac{-4 - 4}{-4 + 3} = rac{-8}{-1} = 8 > 0 \) â positive
["# Solving Inequalities with Positive Ratios: Analyzing ( \frac{-4 - 4}{-4 + 3} = \frac{-8}{-1} = 8 > 0 )", "When solving inequalities involving rational expressions, understanding how negative values behave can clarify both the algebra and the sign of the result. Let’s examine a specific example: For ( x < -3 ), consider ( x = -4 ), and evaluate the expression:", "[\n\frac{-4 - 4}{-4 + 3} = \frac{-8}{-1} = 8 > 0\n]", "### Breaking Down the Calculation Step-by-Step", "1. Substitute ( x = -4 ):\n Plug the value into the expression.", "[\n \frac{-4 - 4}{-4 + 3}\n ]", "2. Simplify numerator:\n [\n -4 - 4 = -8\n ]", "3. Simplify denominator:\n [\n -4 + 3 = -1\n ]", "4. Divide numerator by denominator:\n [\n \frac{-8}{-1} = 8\n ]", "5. Check the result’s sign:\n ( 8 > 0 ), which means the expression evaluates to a positive number.", "---", "### Why Is the Expression Positive When ( x < -3 )?", "The key insight lies in how negative values interact with division:", "- The numerator (-4 - 4 = -8) is negative.\n- The denominator (-4 + 3 = -1) is also negative.\n- Division of two negative numbers yields a positive result:\n [\n \frac{\ ext{negative}}{\ ext{negative}} = \ ext{positive}\n ]", "This matches the calculation that produces (8), a clearly positive value.", "---", "### Why Does This Matter in Inequalities?", "When solving inequalities like ( \frac{x - a}{x - b} > 0 ) (with ( x < -3 )), simply substituting numbers or analyzing intervals helps identify where the expression is positive. Since both numerator and denominator are negative (or both positive) in certain regions, the ratio is positive — a critical insight when determining valid solution sets.", "For ( x < -3 ), expressions involving such ratios maintain positivity unless numerator and denominator differ in sign, but in our example:", "- Numerator: ( x - 4 < -7 \Rightarrow \ ext{negative} )\n- Denominator: ( x + 3 < 0 \Rightarrow \ ext{negative} )\n- Ratio: ( \frac{-\ ext{number}}{-\ ext{number}} = +\ ext{positive} )", "---", "### Summary", "For ( x = -4 ) (a value less than (-3)):", "[\n\frac{-4 - 4}{-4 + 3} = \frac{-8}{-1} = 8 > 0\n]", "This confirms the expression yields a positive result, consistent with how negative numbers divide to produce positive values. Understanding such signs is vital in solving rational inequalities and interpreting inequality solutions correctly.", "---", "### Key Takeaways", "- Two negative values in a fraction produce a positive result.\n- When ( x < -3 ), ( \frac{x - 4}{x + 3} ) is typically positive due to both terms being negative.\n- Substituting concrete values helps visualize and verify abstract expressions.", "Mastering sign analysis in rational expressions allows accurate solutions to inequality problems — especially those involving negative inputs like ( x = -4 ).", "---", "Keywords: rational expressions, positive ratio analysis, negative ( x ), inequality solving, ( \frac{x - a}{x - b} > 0 ), sign of quotient, ( x < -3 ), evaluate expression", "---", "Understanding how negative values combine under division ensures clarity when working with inequalities — turning complex signs into intuitive results."]
