\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0.

["# Solving the Inequality (\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0): A Complete Guide", "Understanding rational inequalities is crucial for mastering algebra and solving real-world problems involving rates, growth, and boundaries. One frequently encountered inequality is:", "[\n\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0\n]", "This article provides a clear, step-by-step guide to solving this inequality, including factoring, analyzing critical points, testing intervals, and interpreting the solution — all optimized for search engines (SEO) to help students, educators, and math enthusiasts find accurate and practical guidance.", "---", "## Why This Inequality Matters", "This rational expression involves a polynomial fraction where the numerator and denominator are quadratic expressions. Solving it involves:", "- Finding zeros of the numerator (where expression equals zero)\n- Finding points where the expression is undefined (undefined or infinite)\n- Analyzing sign changes across intervals\n- Combining results in an accurate, complete solution set", "Mastering such problems strengthens your understanding of domain restrictions, sign charts, and inequality rules — essential tools in advanced math, engineering, and applied sciences.", "---", "## Step 1: Factor the Numerator and Denominator", "Start by factoring both the numerator and denominator to simplify analysis.", "### Numerator: (x^2 - 4x + 3)", "Find two numbers that multiply to 3 and add to -4:\nThese are -1 and -3.\n[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "### Denominator: (x^2 - 5x + 6)", "Factor the quadratic:\nFind two numbers multiplying to 6 and adding to -5: -2 and -3.\n[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "---", "## Step 2: Rewrite the Inequality", "Substitute factored forms into the inequality:", "[\n\frac{(x - 1)(x - 3)}{(x - 2)(x - 3)} \geq 0\n]", "Note: The term (x - 3) appears in both numerator and denominator, suggesting a hole in the graph at (x = 3), but more importantly, this factor will affect domain restrictions.", "---", "## Step 3: Identify Critical Points", "Critical points occur where the expression equals zero or is undefined. These divide the real line into intervals.", "### Zeros (Numerator = 0):\n[\n(x - 1)(x - 3) = 0 \Rightarrow x = 1 \ ext{ or } x = 3\n]", "### Undefined Points (Denominator = 0):\n[\n(x - 2)(x - 3) = 0 \Rightarrow x = 2 \ ext{ or } x = 3\n]", "Even though (x = 3) makes both numerator and denominator zero, it’s excluded due to division by zero. We treat (x = 3) as a vertical asymptote or undefined point.", "---", "## Step 4: Determine the Domain", "Because the denominator is zero at (x = 2) and (x = 3), these values are excluded from the solution set.", "Domain:\n[\nx \in \mathbb{R} \setminus {2, 3}\n]", "---", "## Step 5: Identify Intervals to Test", "The critical points (x = 1), (x = 2), (x = 3), and (x = 3) (repeated) divide the number line into intervals:", "1. ( (-\infty, 1) )\n2. ( (1, 2) )\n3. ( (2, 3) )\n4. ( (3, \infty) )", "Note: (x = 3) is excluded — no value equals 3, but it’s a boundary point to watch due to being a zero and a breach in definedness.", "---", "## Step 6: Analyze Sign of Each Factor in Intervals", "We analyze the sign of each linear factor across intervals:", "| Interval | (x - 1) | (x - 2) | (x - 3) | Expression ( \frac{(x - 1)(x - 3)}{(x - 2)(x - 3)} ) |\n|---------------|-----------|-----------|----------|-------------------------------------------------------|\n| ( (-\infty, 1) ) | – | – | – | (\frac{(-)(-)}{(-)(-)} = \frac{+}{+} = +) |\n| ( (1, 2) ) | + | – | – | (\frac{(+)(-)}{(-)(-)} = \frac{-}{+} = -) |\n| ( (2, 3) ) | + | + | – | (\frac{(+)(-)}{(+)(-)} = \frac{-}{-} = +) |\n| ( (3, \infty) ) | + | + | + | (\frac{(+)(+)}{(+)(+)} = \frac{+}{+} = +) |", "Key observations:", "- At (x = 1): Expression = 0 → included (since inequality is (\geq 0))\n- At (x = 3): Expression undefined → excluded\n- At (x = 2): Denominator zero → excluded (vertical asymptote)\n- The expression alternates sign across each interval because all factors are linear.", "---", "## Step 7: Determine Where Inequality Holds", "We seek where the expression (\geq 0), i.e., positive or zero.", "From the table:", "- Positive in: ((-∞, 1]) and ((2, 3)) and ((3, ∞))\n- Zero at: (x = 1) (excluded if needed for strict inequality, but included here because (\leq 0) allows equality)", "But recall: At (x = 3), expression is undefined — skip this point.", "So the solution intervals are:", "- ((-∞, 1]) — expression ≥ 0\n- ((2, 3)) — expression > 0\n- ((3, ∞)) — expression > 0", "We include (x = 1) because the inequality允许 equality when numerator is zero.", "We exclude (x = 2) and (x = 3), but note (x = 3) lies between the sign changes — no jump discontinuity.", "---", "## Step 8: Write Final Solution in Interval Notation", "Combining intervals where the expression is non-negative:", "[\nx \in (-\infty, 1] \cup (2, 3) \cup (3, \infty)\n]", "Important: (x = 3) is excluded, even though numerator and denominator cancel — because division by zero is undefined.", "---", "## Step 9: Final Answer and Boxed Result", "[\n\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0 \quad \ ext{when} \quad x \in (-\infty, 1] \cup (2, 3) \cup (3, \infty)\n]", "---", "## Bonus Tips for Mastery", "- Always factor completely before analyzing signs.\n- Identify holes (where numerator and denominator share a factor but denominator zero) — here, (x = 3) — but since original expression is undefined, exclude it.\n- Use a sign chart to visualize behavior across critical points.\n- Test a sample point in each interval to confirm sign — not required algebraically but helpful.\n- Remember: (\geq 0) includes zeros but excludes exclusions (denominator zero).", "---", "## Use in Real-World Contexts", "This type of inequality appears in:", "- Electrical engineering when modeling system stability (signs of transfer functions)\n- Economics for break-even analysis involving rational cost-revenue models\n- Biology in growth models where thresholds determine viability", "Understanding such expressions helps predict feasible ranges in modeling.", "---", "## Conclusion", "Solving (\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0) involves factoring, identifying domain exclusions, building a sign chart, and testing intervals. Follow each step methodically — not only to find the answer, but to deepen your algebraic reasoning. With practice, rational inequalities become intuitive, unlocking powerful problem-solving skills across STEM fields.", "---", "SEO Keywords:\n(\frac{x^2 - 4x + 3}{x^2 - 5x + 6} \geq 0), rational inequality, solving rational expressions, step-by-step guide, algebra 2, quadratic inequalities, sign chart, domain analysis, hole in rational function, inequality solution, real number intervals, math tutoring, algebra factoring, exponent rules.", "---", "Start solving rational inequalities with confidence — your path to mathematical mastery begins here."]