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. 3x^2+10x−8≤0

Verified step by step guidance

1. First, identify the polynomial inequality: \(3x^2 + 10x - 8 \leq 0\).
2. Factor the quadratic expression \(3x^2 + 10x - 8\) if possible. Look for two numbers that multiply to \(3 \times (-8) = -24\) and add to \(10\).
3. Once factored, set each factor equal to zero to find the critical points: \(3x^2 + 10x - 8 = 0\).
4. Use the critical points to divide the number line into intervals. Test a point from each interval in the inequality \(3x^2 + 10x - 8 \leq 0\) to determine where the inequality holds true.
5. Express the solution set in interval notation based on the intervals where the inequality is satisfied, and graph the solution set on a real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared 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 on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', while (a, b) excludes both endpoints.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the values that satisfy the inequality. This involves marking the critical points (roots) and shading the appropriate regions based on the intervals where the inequality is true. This graphical representation helps in understanding the solution set's extent and behavior.