Solve each quadratic inequality. Give the solution set in interval notation. 2x²-9x≤18

Verified step by step guidance

1. Start by rewriting the inequality in standard form by moving all terms to one side: \(2x^{2} - 9x \leq 18\) becomes \(2x^{2} - 9x - 18 \leq 0\).
2. Next, factor the quadratic expression \$2x^{2} - 9x - 18\(. To do this, look for two numbers that multiply to \(2 \times (-18) = -36\) and add to \)-9$.
3. Once factored, set each factor equal to zero to find the critical points (roots) of the quadratic equation. These points divide the number line into intervals.
4. Determine the sign of the quadratic expression on each interval by choosing test points from each interval and substituting them back into the factored inequality.
5. Based on the inequality \(\leq 0\), select the intervals where the quadratic expression is less than or equal to zero, and write the solution set in interval notation including the roots where the expression equals zero.

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 its graph.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. Techniques include factoring, completing the square, or using the quadratic formula to find critical points that divide the number line.

Interval Notation and Test Intervals

After finding critical points, the number line is divided into intervals. Test points from each interval in the original inequality to determine where it holds true. The solution set is then expressed in interval notation, showing all x-values that satisfy the inequality.