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. x² ≤ 4x − 2

Verified step by step guidance

1. Rewrite the inequality \(x^2 \leq 4x - 2\) by bringing all terms to one side to set the inequality to zero: \(x^2 - 4x + 2 \leq 0\).
2. Identify the quadratic expression \(x^2 - 4x + 2\) and find its roots by solving the equation \(x^2 - 4x + 2 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-4\), and \(c=2\).
3. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots and then find the exact roots using the quadratic formula.
4. Use the roots to divide the real number line into intervals. Test a value from each interval in the inequality \(x^2 - 4x + 2 \leq 0\) to determine where the inequality holds true.
5. Express the solution set as an interval or union of intervals based on the test results, and graph this solution set on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols (e.g., ≤, ≥, <, >). 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 Finding Critical Points

To solve polynomial inequalities, rewrite the inequality so one side is zero, then factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested to determine where the inequality holds.

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality is true, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a real number line visually represents these intervals, showing included endpoints with closed dots for ≤ or ≥.