Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 4x^2−4x+1≥0

Verified step by step guidance

1. First, recognize that the inequality \(4x^2 - 4x + 1 \geq 0\) is a quadratic inequality.
2. To solve the inequality, start by finding the roots of the quadratic equation \(4x^2 - 4x + 1 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -4\), and \(c = 1\).
3. Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Substitute the values to get \((-4)^2 - 4 \cdot 4 \cdot 1\).
4. Since the discriminant is zero, the quadratic has exactly one real root. Use the quadratic formula to find this root.
5. The solution set for the inequality \(4x^2 - 4x + 1 \geq 0\) is determined by the fact that the quadratic is a perfect square trinomial, which means it is always non-negative. Express the solution set in interval notation and graph it on a real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions that compare a polynomial to a value, typically zero, using inequality symbols such as ≥, ≤, >, or <. To solve these inequalities, one must determine the intervals where the polynomial is either above or below the specified value. This often requires finding the roots of the polynomial and testing intervals between these roots.

Graphing on a Number Line

Graphing the solution set of a polynomial inequality on a number line visually represents the intervals where the inequality holds true. Solutions are typically marked with open or closed circles to indicate whether endpoints are included (closed) or excluded (open). This graphical representation helps in understanding the range of values that satisfy the inequality.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is essential for clearly expressing the solution set of inequalities.