oxed{210}Question: Find all real numbers $ x $ such that $ rac{x+3}{x-2} < 4 $, where $ x

["Title: Solve the Inequality: ( \frac{x+3}{x-2} < 4 ) — Step-by-Step Solution", "Meta Description:\nLearn how to solve the inequality ( \frac{x+3}{x-2} < 4 ) step by step. Find all real values of $ x $ satisfying the inequality, including domain considerations and critique of key steps.", "---", "## Introduction", "Solving rational inequalities like ( \frac{x+3}{x-2} < 4 ) is a crucial skill in algebra, helping develop logical thinking and precision. In this detailed guide, we explain how to find all real numbers $ x $ such that this inequality holds, breaking down every step clearly with a focus on correctness and conceptual understanding.", "---", "## Understanding the Problem", "We are solving:", "[\n\frac{x+3}{x-2} < 4\n]", "This inequality compares a rational expression to a constant (4). The key challenge lies in handling the fraction carefully — especially the domain restriction (where the denominator is not zero) and the sign changes when multiplying across expressions with variable expressions.", "---", "## Step 1: Find Domain Restrictions", "Before solving, identify values of $ x $ that make the denominator zero, since division by zero is undefined.", "[\nx - 2 = 0 \Rightarrow x = 2\n]", "So, $ x = 2 $ is excluded from the domain. This splits the real number line into two intervals:", "[\n(-\infty, 2) \quad \ ext{and} \quad (2, \infty)\n]", "We will solve the inequality separately on these intervals.", "---", "## Step 2: Rewrite the Inequality", "Start with:", "[\n\frac{x+3}{x-2} < 4\n]", "Subtract 4 from both sides to bring all terms to one side:", "[\n\frac{x+3}{x-2} - 4 < 0\n]", "Now express 4 as a fraction with denominator $ x - 2 $:", "[\n\frac{x+3}{x-2} - \frac{4(x-2)}{x-2} = \frac{x+3 - 4x + 8}{x-2} = \frac{-3x + 11}{x-2}\n]", "So the inequality becomes:", "[\n\frac{-3x + 11}{x - 2} < 0\n]", "We can factor numerator by factoring out $-1$ for simplicity:", "[\n\frac{-(3x - 11)}{x - 2} < 0\n]", "Multiplying numerator and denominator by $-1$ reverses the inequality:", "[\n\frac{3x - 11}{x - 2} > 0\n]", "We now solve:", "[\n\frac{3x - 11}{x - 2} > 0\n]", "---", "## Step 3: Find Critical Points", "Set numerator and denominator equal to zero:", "- $ 3x - 11 = 0 \Rightarrow x = \frac{11}{3} \approx 3.67 $\n- $ x - 2 = 0 \Rightarrow x = 2 $", "These points divide the real line into three intervals:", "1. $ (-\infty, 2) $\n2. $ (2, \frac{11}{3}) $\n3. $ \left(\frac{11}{3}, \infty\right) $", "---", "## Step 4: Analyze Sign of the Expression", "We analyze the sign of $ \frac{3x - 11}{x - 2} $ in each interval:", "| Interval | Choose Test Value | Numerator $ 3x - 11 $ | Denominator $ x - 2 $ | Sign of Fraction |\n|--------------------|-------------------|------------------------|------------------------|------------------|\n| $ (-\infty, 2) $ | $ x = 0 $ | $ -11 $ (–) | $ -2 $ (–) | Positive (+) |\n| $ (2, \frac{11}{3}) $ | $ x = 3 $ | $ 3(3)-11 = -2 $ (–) | $ 1 $ (+) | Negative (–) |\n| $ (\frac{11}{3}, \infty) $ | $ x = 4 $ | $ 12 - 11 = 1 $ (+) | $ 2 $ (+) | Positive (+) |", "The expression is positive in $ (-\infty, 2) $ and $ \left(\frac{11}{3}, \infty\right) $.", "---", "## Step 5: Apply Strict Inequality", "Recall we have:", "[\n\frac{3x - 11}{x - 2} > 0\n]", "We exclude $ x = 2 $ entirely (undefined), and since we require strict inequality, endpoints where numerator is zero (i.e., $ x = \frac{11}{3} $) are not included (it makes numerator 0, so expression = 0), which does not satisfy $ > 0 $.", "Therefore, solution is:", "[\nx \in (-\infty, 2) \cup \left(\frac{11}{3}, \infty\right)\n]", "---", "## Step 6: Final Answer and Domain Check", "Thus, all real numbers $ x $ satisfying $ \frac{x+3}{x-2} < 4 $ are:", "[\nx < 2 \quad \ ext{or} \quad x > \frac{11}{3}\n]", "Expressed in boxed form:", "[\n\boxed{x < 2 \quad \ ext{or} \quad x > \frac{11}{3}}\n]", "---", "## Additional Notes", "- The excluded point $ x = 2 $ correctly avoids division by zero.\n- The sign analysis confirms the solution aligns with the rational expression’s behavior.\n- This inequality type—subtracting a constant and analyzing sign—is a common and powerful technique in algebra.", "---", "## Conclusion", "Solving ( \frac{x+3}{x-2} < 4 ) involves careful domain analysis, algebraic manipulation, and interval testing. By rewriting as a single rational expression and determining where it is positive, we find the precise set of real solutions. This method applies broadly to rational inequalities and strengthens logical and computational fluency.", "---", "Keywords: rational inequality, solve ( \frac{x+3}{x-2} < 4 ), step-by-step algebra, domain exclusion, sign analysis, real number solutions, solving rational expressions", "Related Articles:\n- How to Solve ( \frac{x}{x-1} > 2 )\n- Complete Guide to Inequalities Involving Rational Expressions\n- Mastering Critical Points in Algebraic Inequalities", "---", "Optimized for search: approaches to solve ( \frac{x+3}{x-2} < 4 ), real solutions, inequality analysis, step-by-step guide."]