For $ x < 2 $, say $ x = 0 $: numerator $ -3(0) + 11 = 11 > 0 $, denominator $ 0 - 2 = -2 < 0 $ → negative.

["Understanding Inequalities and Expressions: Analyzing the Sign of a Rational Function for ( x < 2 )", "When evaluating rational expressions, understanding the sign of the numerator and denominator is crucial, especially when comparing values less than a critical threshold. Take the expression:", "[\n\frac{-3x + 11}{x - 2}\n]", "We focus on the case where ( x < 2 ), particularly presenting the key example: when ( x = 0 ).", "### Evaluating the Expression at ( x = 0 )", "Substituting ( x = 0 ):", "- Numerator:\n [\n -3(0) + 11 = 0 + 11 = 11\n ]\n This evaluates to a positive value: ( 11 > 0 ).", "- Denominator:\n [\n 0 - 2 = -2\n ]\n This is clearly negative: ( -2 < 0 ).", "Since we have a positive numerator and a negative denominator, the entire fraction becomes:", "[\n\frac{+}{–} = -\ ext{(negative number)}\n]", "Thus,\n[\n\frac{-3(0) + 11}{0 - 2} = \frac{11}{-2} = -5.5 < 0\n]", "Which confirms the result: for ( x = 0 ) (and all ( x < 2 )), the value of the expression is negative.", "### Why This Matters: Sign Analysis in Rational Functions", "Understanding the signs of both numerator and denominator helps determine the overall sign of a rational function, which has practical applications in:", "- Solving inequalities with fractions\n- Predicting changes in behavior across intervals\n- Graphing rational functions by identifying zeroes and asymptotes", "For all ( x < 2 ), the numerator (-3x + 11) stays positive because it increases as ( x ) decreases from 2 toward negative infinity and starts at 11 when ( x = 0 ). Meanwhile, the denominator ( x - 2 ) remains negative throughout the interval, since ( x < 2 ) implies values less than 2.", "When a positive quantity divides into a negative one, the result is negative — explaining why the expression is always negative for all ( x < 2 ).", "### Key Takeaways", "- When ( x < 2 ), ( x - 2 < 0 ) (negative).\n- The numerator, (-3x + 11), is positive for all ( x < 2 ), starting high when ( x = 0 ).\n- Dividing a positive by a negative always yields a negative result.", "Thus, for ( x < 2 ),\n[\n\frac{-3x + 11}{x - 2} < 0\n]", "### Practical Tip", "Always break down rational expressions into numerator and denominator sign tests before solving inequalities or simplifying. This method works for any rational function and helps visualize trends quickly.", "---", "Keywords: rational expression sign analysis, evaluate numerator and denominator, negative rational function ( x < 2 ), inequality sign rules, fraction sign chart, ( \frac{-3x + 11}{x - 2} < 0 ), understanding sign changes in rational functions."]