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. 4x² + 1 ≥ 4x

Verified step by step guidance

1. Rewrite the inequality to have zero on one side by subtracting \(4x\) from both sides: \(4x^2 + 1 - 4x \geq 0\).
2. Rearrange the terms to standard quadratic form: \(4x^2 - 4x + 1 \geq 0\).
3. Identify the quadratic expression \(4x^2 - 4x + 1\) and find its roots by solving the 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 \(c=1\).
4. Determine the nature of the roots by calculating the discriminant \(\Delta = b^2 - 4ac\). This will tell you if the quadratic touches or crosses the x-axis.
5. Use the roots (if any) to divide the number line into intervals, then test a value from each interval in the inequality \(4x^2 - 4x + 1 \geq 0\) to determine where the inequality holds true. Express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring and Solving Quadratic Equations

To solve polynomial inequalities, especially quadratics like 4x² + 1 ≥ 4x, it is helpful to rewrite the inequality in standard form and factor or use the quadratic formula. This helps identify critical points where the expression equals zero, which divide the number line into intervals for testing.

Interval Notation and Graphing Solution Sets

After determining where the polynomial is positive or negative, solutions are expressed using interval notation, which concisely represents sets of numbers. Graphing on a number line visually shows these intervals, indicating where the inequality holds true with open or closed circles depending on strict or inclusive inequalities.