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+1≥4x

Verified step by step guidance

1. Rewrite the inequality in standard form: $4x^2 - 4x + 1 \geq 0$.
2. Identify the corresponding quadratic equation: $4x^2 - 4x + 1 = 0$.
3. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots, where $a = 4$, $b = -4$, and $c = 1$.
4. Determine the intervals to test by using the roots found in the previous step to divide the number line.
5. Test each interval in the inequality $4x^2 - 4x + 1 \geq 0$ to determine where the inequality holds true, and express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions that compare a polynomial to a value using inequality symbols (e.g., ≥, ≤, >, <). To solve these inequalities, one typically finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses brackets [ ] to include endpoints and parentheses ( ) to exclude them. For example, the interval [a, b) includes 'a' but not 'b', indicating all numbers from 'a' to 'b', including 'a' and excluding 'b'.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the solutions to an inequality. Each solution is marked with a solid dot for included endpoints and an open dot for excluded endpoints. This graphical representation helps in understanding the range of values that satisfy the inequality, making it easier to interpret the results.