For $ x < -3 $, test $ x = -4 $: $ \frac{-12}{-1} = 12 > 0 $.

["Understanding Division of Negative Numbers: Testing with ( x = -4 ) for ( x < -3 )", "When working with negative numbers, especially in inequality testing, it’s essential to understand how signs affect the outcome of mathematical operations. This foundational concept is vital in algebra, model building, and real-world calculations. One common test that clarifies the behavior of signed expressions involves substituting a value like ( x = -4 ), particularly when ( x < -3 ).", "### The Expression in Focus: ( \frac{-12}{-1} = 12 > 0 )", "Let’s analyze a specific example to illustrate the rule: for all values where ( x < -3 ), test ( x = -4 ) and compute ( \frac{-12}{x} ).", "Substitute ( x = -4 ):", "[\n\frac{-12}{-4} = 3\n]", "Since ( 3 > 0 ), this confirms the expected result: a negative numerator divided by a negative denominator yields a positive quotient.", "### Why Does This Happen?", "Key Rule: The quotient of two negative numbers is positive. By this rule:", "- Negative ÷ Negative = Positive", "Thus, ( \frac{-12}{-4} = 3 ), clearly showing positivity.", "Because ( -4 < -3 ), this example validates not only the arithmetic but also reinforces the inequality condition ( x < -3 ), ensuring the denominator remains negative — crucial for producing a positive result.", "### Broader Implications in Algebra", "Testing specific values like ( x = -4 ) helps verify whether an expression behaves as expected across intervals. It builds confidence in solving equations, simplifying rational expressions, and analyzing functions. Understanding that negative divided by negative equals positive enables students and educators to avoid common sign errors and interpret expressions correctly in inequalities and inequalities solving.", "### Conclusion", "Testing ( x = -4 ) (all values less than -3) in expressions such as ( \frac{-12}{x} ) confirms that negative numerators divided by negative denominators yield positive results. This fundamental sign rule is essential for mastering algebraic reasoning, solving inequalities, and applying math effectively in scientific and engineering contexts.", "Keywords: ( x < -3 ), negative numbers, division rules, test value ( x = -4 ), negative divided by negative, algebra, negative reciprocal, sign of quotient, mathematical reasoning.", "---", "Summary:\nTesting ( x = -4 ) confirms that ( \frac{-12}{-1} = 12 > 0 ) for ( x < -3 ). This demonstrates how dividing two negative numbers produces a positive result — a core principle in algebra and rational expressions. Always test values in the relevant domain to verify expected sign behavior."]