Now analyze the sign of \( rac{x - 4}{x + 3} \)

["# Analyzing the Sign of ( \frac{x - 4}{x + 3} ): A Comprehensive Guide", "When working with rational expressions like ( \frac{x - 4}{x + 3} ), understanding the sign of the expression across different intervals of ( x ) is crucial for solving inequalities, graphing rational functions, and interpreting real-world data. In this article, we’ll walk through a step-by-step analysis of the sign of ( \frac{x - 4}{x + 3} ), helping you determine where it’s positive, negative, or undefined.", "---", "## What Does the Sign Mean?", "The sign (or sign analysis) of a rational expression tells us whether the expression is greater than zero (positive), less than zero (negative), or zero—across intervals defined by the expression’s critical points: values that make the numerator zero or the denominator zero.", "For ( \frac{x - 4}{x + 3} ):", "- The numerator ( x - 4 = 0 ) when ( x = 4 ), creating a zero of the expression.\n- The denominator ( x + 3 = 0 ) when ( x = -3 ), causing a vertical asymptote and a discontinuity, meaning the expression is undefined here.", "These two points divide the number line into intervals we’ll analyze.", "---", "## Step-by-Step Sign Analysis", "### Step 1: Identify critical points\nThe critical points are:\n- ( x = -3 ) (denominator zero → undefined)\n- ( x = 4 ) (numerator zero → zero of the function)", "These points split the number line into three intervals:\n1. ( (-\infty, -3) )\n2. ( (-3, 4) )\n3. ( (4, \infty) )", "---", "### Step 2: Choose test points in each interval", "Pick a number from each interval and substitute it into:", "[\nf(x) = \frac{x - 4}{x + 3}\n]", "| Interval | Test Point | ( x - 4 ) | ( x + 3 ) | Sign of ( f(x) ) |\n|---------------------|------------|-------------|-------------|---------------------|\n| ( (-\infty, -3) ) | ( x = -4 )| (-8) | (-1) | ( + ) (positive) |\n| ( (-3, 4) ) | ( x = 0 ) | (-4) | (+3) | ( - ) (negative) |\n| ( (4, \infty) ) | ( x = 5 ) | (+1) | (+8) | ( + ) (positive) |", "---", "### Step 3: Analyze results", "- In ( (-\infty, -3) ): both numerator and denominator negative → positive (+)\n- In ( (-3, 4) ): numerator negative, denominator positive → negative (−)\n- In ( (4, \infty) ): both numerator and denominator positive → positive (+)", "---", "### Step 4: Behavior at critical points", "- At ( x = -3 ): The expression is undefined. The function has a vertical asymptote.\n- At ( x = 4 ): The numerator is zero, denominator nonzero → function equals zero. This is a zero crossing.", "Note: The sign does not include ( x = -3 ), since the function is undefined there.", "---", "## Summary of Signs", "| Interval | Sign of ( \frac{x - 4}{x + 3} ) |\n|----------------------|------------------------------------|\n| ( (-\infty, -3) ) | + (positive) |\n| ( (-3, 4) ) | − (negative) |\n| ( (4, \infty) ) | + (positive) |", "---", "## Why Is This Analysis Important?", "Understanding the sign helps in:", "- Solving inequalities (e.g., ( \frac{x - 4}{x + 3} > 0 ))\n- Sketching the graph (locating zeros and asymptotes)\n- Interpreting physical or economic models that use rational functions", "---", "## Final Thoughts", "Analyzing the sign of rational expressions like ( \frac{x - 4}{x + 3} ) involves identifying zeros and undefined points, selecting test values, and checking signs in each interval. This method ensures accuracy and clarity when working with rational functions—essential for students, educators, and professionals alike.", "If you want to go further, explore how sign changes reflect continuity, limits, and real-world applications like rate problems or mixture models.", "---", "Key Takeaway:\nThe expression ( \frac{x - 4}{x + 3} ) is positive when ( x < -3 ) or ( x > 4 ), negative when ( -3 < x < 4 ), undefined at ( x = -3 ), and zero at ( x = 4 ).", "---", "Use this guide to confidently analyze rational functions and their signs—mastering a key skill in algebra!"]