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. 3x²+10x−8≤0

Verified step by step guidance

1. Start by writing down the inequality: \$3x^{2} + 10x - 8 \leq 0$.
2. To solve the inequality, first find the roots of the corresponding quadratic equation \$3x^{2} + 10x - 8 = 0\( by using the quadratic formula: \)x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, where \)a=3\(, \)b=10\(, and \)c=-8$.
3. Calculate the discriminant \(\Delta = b^{2} - 4ac = 10^{2} - 4 \times 3 \times (-8)\) to determine the nature of the roots.
4. Find the two roots by substituting the values into the quadratic formula. These roots will divide the real number line into intervals.
5. Test a value from each interval in the original inequality \$3x^{2} + 10x - 8 \leq 0$ to determine which intervals satisfy the inequality, then express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or 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 Quadratic Polynomials

Factoring is the process of rewriting a quadratic polynomial as a product of two binomials. This helps identify the roots (zeros) of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial satisfies the inequality, using open or closed dots to indicate whether endpoints are included.