Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. 2x^3 - 7x^2 ≥ 3 - 8x

Verified step by step guidance

1. Step 1: First, we need to rearrange the inequality to have all terms on one side. This can be done by adding 8x to both sides and subtracting 3 from both sides. This gives us the inequality 2x^3 - 7x^2 + 8x - 3 ≥ 0.
2. Step 2: Next, we need to find the roots of the polynomial 2x^3 - 7x^2 + 8x - 3 = 0. This can be done by factoring the polynomial or using the rational root theorem.
3. Step 3: Once we have the roots, we can use them to divide the number line into intervals. We then pick a test point from each interval and substitute it into the inequality. If the inequality is true for the test point, then all numbers in that interval are solutions to the inequality.
4. Step 4: We repeat the process for each interval. The union of all intervals for which the inequality is true is the solution set to the inequality.
5. Step 5: Finally, we express the solution set in interval notation. This is done by writing each interval as (a, b), [a, b), (a, b], or [a, b] depending on whether the endpoints are included in the solution set. The union of intervals is represented by the symbol ∪.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions that compare a polynomial to a value using inequality symbols (e.g., ≥, ≤, >, <). To solve these inequalities, one typically finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses brackets and parentheses to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, [a, b] includes both a and b, while (a, b) does not include them.

Testing Intervals

After identifying the roots of a polynomial inequality, the next step is to test the sign of the polynomial in the intervals created by these roots. By selecting test points from each interval, one can determine whether the polynomial is positive or negative in those intervals, which helps in finding the solution set for the inequality.