Solve each polynomial inequality. Give the solution set in interval notation. 2x³ - 7x² ≥ 3 - 8x

Verified step by step guidance

1. First, rewrite the inequality so that all terms are on one side, setting the inequality to be greater than or equal to zero. This means subtracting 3 and adding 8x to both sides: \$2x^{3} - 7x^{2} - 3 + 8x \geq 0$.
2. Next, combine like terms to simplify the expression: \$2x^{3} - 7x^{2} + 8x - 3 \geq 0$.
3. Now, factor the polynomial expression on the left side if possible. Start by looking for rational roots using the Rational Root Theorem or by trial, then use polynomial division or synthetic division to factor the cubic polynomial into linear and/or quadratic factors.
4. After factoring, identify the critical points by setting each factor equal to zero and solving for \(x\). These points divide the number line into intervals.
5. Finally, test a value from each interval in the original inequality to determine where the inequality holds true. Use this information to write the solution set in interval notation, including endpoints where the inequality is 'greater than or equal to' zero.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value or polynomial 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.

Rearranging and Simplifying Inequalities

To solve polynomial inequalities, first rewrite the inequality so that one side is zero by moving all terms to one side. This simplification allows factoring or other methods to find critical points where the polynomial equals zero, which are essential for testing intervals.

Sign Analysis and Interval Notation

After finding the roots of the polynomial, use sign analysis to determine where the polynomial is positive or negative by testing values in intervals defined by the roots. The solution set is then expressed in interval notation, which concisely represents all values satisfying the inequality.