- 12y + 4y^2 \leq y^2 - 4y + 4

["Understanding the Inequality: 12y + 4y² ≤ y² - 4y + 4", "Solving inequalities like 12y + 4y² ≤ y² - 4y + 4 may seem challenging at first, but with the right approach, you can determine the range of values for y that satisfy this condition. This article breaks down the inequality step by step to help you understand how to solve it and interpret its solutions effectively for both math learning and real-world applications.", "---", "### What Is the Inequality?", "We begin with:\n12y + 4y² ≤ y² - 4y + 4", "This is a quadratic inequality involving a linear and a quadratic expression. Our goal is to find all real numbers y for which this inequality holds true.", "---", "### Step 1: Bring All Terms to One Side", "To simplify, subtract y² − 4y + 4 from both sides:", "4y² + 12y − (y² − 4y + 4) ≤ 0", "Simplify:\n4y² + 12y − y² + 4y − 4 ≤ 0\n→ 3y² + 16y − 4 ≤ 0", "Now, the inequality becomes:\n3y² + 16y − 4 ≤ 0", "---", "### Step 2: Solve the Corresponding Quadratic Equation", "To find the boundary points, solve the related equation:\n3y² + 16y − 4 = 0", "Use the quadratic formula:\n[\ny = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere a = 3, b = 16, c = -4.", "Calculate the discriminant:\n[\n\Delta = 16^2 - 4(3)(-4) = 256 + 48 = 304\n]\nSo,\n[\ny = \frac{-16 \pm \sqrt{304}}{6}\n]", "Simplify √304:\n√304 = √(16 × 19) = 4√19, so:\n[\ny = \frac{-16 \pm 4\sqrt{19}}{6} = \frac{-8 \pm 2\sqrt{19}}{3}\n]", "Thus, the two real roots are:\n[\ny_1 = \frac{-8 - 2\sqrt{19}}{3}, \quad y_2 = \frac{-8 + 2\sqrt{19}}{3}\n]", "Approximate √19 ≈ 4.3589, so:\n[\ny_1 \approx \frac{-8 - 8.7178}{3} = \frac{-16.7178}{3} \approx -5.5726\n]\n[\ny_2 \approx \frac{-8 + 8.7178}{3} = \frac{0.7178}{3} \approx 0.2393\n]", "---", "### Step 3: Analyze the Quadratic Expression", "The quadratic 3y² + 16y − 4 opens upward because the coefficient of y² (i.e., 3) is positive.", "Since the parabola opens upward, the expression is less than or equal to zero between the two real roots.", "Thus, the solution to 3y² + 16y − 4 ≤ 0 is:\n[\n\frac{-8 - 2\sqrt{19}}{3} \leq y \leq \frac{-8 + 2\sqrt{19}}{3}\n]", "Or approximately:\n[\n-5.57 ≤ y ≤ 0.24\n]", "---", "### Step 4: Final Answer & Interpretation", "We write the solution set as:\n[\n\boxed{ \frac{-8 - 2\sqrt{19}}{3} \leq y \leq \frac{-8 + 2\sqrt{19}}{3} }\n]", "This means the inequality 12y + 4y² ≤ y² − 4y + 4 holds true exclusively between the two roots found. Outside this interval, the quadratic expression exceeds zero.", "---", "### Why Does This Matter?", "Understanding such inequalities plays a vital role in fields like algebraic modeling, economics, engineering, and data science, where constraints and feasible regions must be analyzed. For instance, when optimizing a cost or profit function subject to inequalities, solving such expressions narrows valid solution intervals.", "---", "Key takeaways:\n- Move all terms to one side to simplify.\n- Use the quadratic formula to find breakpoints.\n- The direction of the inequality determines the solution range.\n- The coefficient of y² determines whether the parabola opens up or down.", "Mastering quadratic inequalities strengthens problem-solving skills essential in math and beyond!", "---", "For further reading:\n- How to solve quadratic inequalities generally\n- Graphing quadratic functions and testing intervals\n- Applications of inequalities in real-world modeling", "---", "Keywords: 12y + 4y² ≤ y² - 4y + 4, quadratic inequality, solving inequalities, algebra, quadratic equation, inequality solution, real numbers, mathematical analysis"]