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. 2x²+x<15

Verified step by step guidance

1. Start by rewriting the inequality so that one side is zero: subtract 15 from both sides to get \(2x^{2} + x - 15 < 0\).
2. Next, factor the quadratic expression \(2x^{2} + x - 15\) if possible. Look for two numbers that multiply to \(2 \times (-15) = -30\) and add to \(1\) (the coefficient of \(x\)).
3. Once factored, write the inequality as a product of two binomials less than zero, for example, \((ax + b)(cx + d) < 0\).
4. Determine 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 see if it satisfies the inequality. Use this to identify which intervals are part of the solution set, then express the solution in interval notation and graph it on the number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols like <, >, ≤, or ≥. 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 in standard form and 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 to determine where the inequality holds.

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality is true, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a real number line visually represents these intervals, showing open or closed endpoints depending on the inequality.