Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 2x²-9x≤18

Verified step by step guidance

1. First, rewrite the inequality in standard form by moving all terms to one side: \$2x^{2} - 9x - 18 \leq 0$.
2. Next, factor the quadratic expression \$2x^{2} - 9x - 18\( if possible, or use the quadratic formula to find its roots. The quadratic formula is given by \)x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, where \)a=2\(, \)b=-9\(, and \)c=-18$.
3. Calculate the discriminant \(\Delta = b^{2} - 4ac\) to determine the nature of the roots. If \(\Delta > 0\), there are two distinct real roots; if \(\Delta = 0\), one real root; if \(\Delta < 0\), no real roots.
4. Use the roots found to divide the number line into intervals. Test a value from each interval in the inequality \$2x^{2} - 9x - 18 \leq 0$ to determine where the inequality holds true.
5. Finally, write the solution set in interval notation, including the roots where the inequality is equal to zero (since it is a 'less than or equal to' inequality).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of x-values that satisfy the inequality, often by analyzing the related quadratic equation and testing intervals.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. This helps find critical points (roots) that divide the number line into intervals for testing the inequality.

Interval Notation and Testing Intervals

After finding the roots, the number line is split into intervals. Each interval is tested in the original inequality to determine if it satisfies the condition. The solution set is then expressed using interval notation to clearly show all valid x-values.