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. 9x²−6x+1<0

Verified step by step guidance

1. Start by identifying the polynomial inequality: \$9x^2 - 6x + 1 < 0$.
2. Recognize that this is a quadratic inequality. To solve it, first find the roots of the corresponding quadratic equation \$9x^2 - 6x + 1 = 0\( by using the quadratic formula: \)x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\(, where \)a=9\(, \)b=-6\(, and \)c=1$.
3. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots. This will help you understand if the quadratic crosses the x-axis or not.
4. Use the roots (if any) to divide the real number line into intervals. Test a value from each interval in the original inequality \$9x^2 - 6x + 1 < 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 then graph this solution set on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like <, >, ≤, or ≥. 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 Quadratic Formula

To solve polynomial inequalities, especially quadratics, you often need to find the roots by factoring or using the quadratic formula. These roots divide the number line into intervals where the polynomial's sign can be tested to determine where the inequality holds.

Interval Notation and Graphing on a Number Line

After determining the solution intervals, expressing them in interval notation provides a concise way to represent all values satisfying the inequality. Graphing these intervals on a real number line visually shows the solution set, indicating whether endpoints are included or excluded.