Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 2x² - 9x ≥ 18

Verified step by step guidance

1. Start by rewriting the inequality so that one side is zero: subtract 18 from both sides to get \$2x^{2} - 9x - 18 \geq 0$.
2. Next, factor the quadratic expression \$2x^{2} - 9x - 18\(. Look for two numbers that multiply to \)2 \times (-18) = -36\( and add to \)-9$ to help with factoring by grouping.
3. Once factored, set each factor equal to zero to find the critical points (roots) of the quadratic. These points divide the number line into intervals.
4. Determine the sign of the quadratic expression on each interval by choosing a test point from each interval and substituting it back into the factored form.
5. Based on the inequality \(\geq 0\), select the intervals where the expression is positive or zero, and write the solution set in interval notation including the critical points where the expression equals zero.

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the inequality true. This typically requires rewriting the inequality in standard form, factoring or using the quadratic formula to find critical points, and then testing intervals to determine where the inequality holds.

Factoring Quadratic Expressions

Factoring is the process of expressing a quadratic expression as a product of two binomials. It helps identify the roots of the quadratic equation, which are essential for determining the intervals to test in an inequality. For example, factoring 2x² - 9x - 18 helps find critical points.

Interval Notation

Interval notation is a concise way to represent sets of real numbers, especially solutions to inequalities. It uses parentheses and brackets to indicate open or closed intervals, respectively, and unions to combine multiple intervals, providing a clear description of the solution set.