Find all solutions to the inequality \( w^2 - 4w + 3 < 0 \).
["# Solve the Inequality ( w^2 - 4w + 3 < 0 ) – All Solutions Explained", "Inequalities are fundamental in algebra, helping students and math enthusiasts understand where expressions are positive, negative, or zero. One commonly studied inequality is:", "[\nw^2 - 4w + 3 < 0\n]", "In this SEO-optimized article, we’ll walk through step-by-step solutions to find all values of ( w ) that satisfy this inequality—plus essential explanations to reinforce your understanding.", "---", "## Step-by-Step Solution to ( w^2 - 4w + 3 < 0 )", "### 1. Factor the Quadratic Expression", "The first step is factoring the quadratic:", "[\nw^2 - 4w + 3 = (w - 1)(w - 3)\n]", "This factorization is easy using basic quadratic formulas or trial and error.", "---", "### 2. Identify Critical Points", "Set the expression equal to zero to find the critical points:", "[\n(w - 1)(w - 3) = 0\n]", "This gives:", "[\nw = 1 \quad \ ext{and} \quad w = 3\n]", "These divide the number line into three intervals:", "- ( (-\infty, 1) )\n- ( (1, 3) )\n- ( (3, \infty) )", "---", "### 3. Test Each Interval", "We test a value from each interval in the original inequality ( (w - 1)(w - 3) < 0 ):", "- For ( w = 0 ) (in ( (-\infty, 1) )):\n ( (0 - 1)(0 - 3) = (-1)(-3) = 3 > 0 )\n Not valid (satisfies ( > 0 ), not ( < 0 ))", "- For ( w = 2 ) (in ( (1, 3) )):\n ( (2 - 1)(2 - 3) = (1)(-1) = -1 < 0 )\n Valid! This interval satisfies the inequality.", "- For ( w = 4 ) (in ( (3, \infty) )):\n ( (4 - 1)(4 - 3) = (3)(1) = 3 > 0 )\n Not valid.", "---", "### 4. Include Boundaries?", "Since the inequality is strict (( < 0 )), equality never holds. Therefore:", "- ( w = 1 ) and ( w = 3 ) are not included.", "---", "## Final Answer", "The inequality ( w^2 - 4w + 3 < 0 ) holds true only when:", "[\n1 < w < 3\n]", "All real numbers ( w ) strictly between 1 and 3 satisfy the inequality.", "---", "## Why This Matters (Educational Takeaway)", "This inequality illustrates the core idea: a quadratic expression changes sign at its real roots, and testing intervals between critical points reveals where the expression is positive or negative. Knowing how to solve such inequalities strengthens foundational algebraic reasoning, essential for calculus, physics modeling, and engineering applications.", "---", "## Tips for Solving Similar Inequalities", "- Always factor first if possible.\n- Find roots carefully—they partition the number line.\n- Test one value per interval to determine the sign.\n- Remember whether the inequality is ( < ), ( > ), ( \leq ), or ( \geq )—this affects inclusion of boundary points.\n- Use a sign chart for clarity.", "---", "## Related Keywords (for SEO optimization)", "- Solve inequality ( w^2 - 4w + 3 < 0\n- Find all solutions to ( w^2 - 4w + 3 < 0 )\n- Quadratic inequality solutions step-by-step\n- Critical points and sign analysis\n- Where is ( w^2 - 4w + 3 < 0 ) satisfied?\n- How to solve ( (w - 1)(w - 3) < 0 )", "---", "### Summary Table", "| Interval | Test Value | Sign of ( (w - 1)(w - 3) ) | Satisfies Inequality? |\n|----------------|------------|------------------------------|------------------------|\n| ( (-\infty, 1) ) | 0 | ( + ) | No |\n| ( (1, 3) ) | 2 | ( - ) | Yes |\n| ( (3, \infty) ) | 4 | ( + ) | No |", "---", "Start solving inequalities confidently—mastering this pattern opens doors to advanced math!\nFor more tutorials, explore quadratic inequalities, sign charts, and critical point analysis.", "---", "Keywords: solve ( w^2 - 4w + 3 < 0 ), find all solutions, quadratic inequality, interval testing, algebra tips"]
