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. $5x\le 2-3x^2$

Verified step by step guidance

1. Rewrite the inequality to have zero on one side by adding \(3x^2\) to both sides: \(5x + 3x^2 \leq 2\).
2. Bring all terms to one side to set the inequality to zero: \(3x^2 + 5x - 2 \leq 0\).
3. Factor the quadratic expression \(3x^2 + 5x - 2\) if possible, or use the quadratic formula to find its roots.
4. Determine the intervals defined by the roots and test a value from each interval in the inequality \(3x^2 + 5x - 2 \leq 0\) to see where it holds true.
5. Express the solution set as an interval or union of intervals based on the test results and graph this solution on the real number line.

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 and Finding Critical Points

To solve polynomial inequalities, first rewrite the inequality so one side is zero, then factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested.

Interval Notation and Graphing Solution Sets

After determining where the polynomial is positive or negative, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a number line visually represents these intervals, showing included or excluded endpoints.