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. 6x²+x>1

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \$6x^{2} + x - 1 > 0$.
2. Factor the quadratic expression \$6x^{2} + x - 1\( if possible, or use the quadratic formula to find its roots. The quadratic formula is given by \)x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, where \)a=6\(, \)b=1\(, and \)c=-1$.
3. Identify the critical points (roots) from the previous step. These points divide the real number line into intervals.
4. Test a value from each interval in the inequality \$6x^{2} + x - 1 > 0$ to determine whether the inequality holds in that interval.
5. Based on the test results, write the solution set as a union of intervals where the inequality is true, and express it in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to 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 Finding Critical Points

To solve polynomial inequalities, it is essential to rewrite the inequality in standard form and 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.

Interval Notation and Graphing Solution Sets

After determining where the polynomial is positive or negative, the solution set is expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing on a number line visually shows these intervals, using open or closed circles to indicate whether endpoints are included.