Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. 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 the right side from 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. Once factored, identify the critical points by setting each factor equal to zero and solving for \(x\). These points divide the number line into intervals.
5. Test a value from each interval in the original inequality to determine whether the inequality holds in that interval. Use this information to write the solution set in interval notation, including endpoints where the inequality is 'greater than or equal to' (≥).

Key Concepts

Polynomial Inequalities

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

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials or factors. This step is crucial for solving inequalities because it helps identify the roots or zeros of the polynomial, which divide the number line into intervals to test for the inequality.

Interval Notation and Sign Analysis

Interval notation expresses the solution set of inequalities using intervals on the real number line. Sign analysis involves testing values from each interval determined by the roots to see where the polynomial satisfies the inequality, allowing the correct intervals to be included in the solution.