Solve each inequality in Exercises 86–91 using a graphing utility. x² + 3x - 10 > 0

Verified step by step guidance

1. Rewrite the inequality to understand the expression clearly: \(x^{2} + 3x - 10 > 0\).
2. Find the roots of the quadratic equation \(x^{2} + 3x - 10 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=3\), and \(c=-10\).
3. Calculate the discriminant \(\Delta = b^{2} - 4ac\) to determine the nature of the roots.
4. Use the roots found to divide the number line into intervals. Test a value from each interval in the inequality \(x^{2} + 3x - 10 > 0\) to determine where the inequality holds true.
5. Express the solution set as intervals where the quadratic expression is greater than zero, based on the test results from the intervals.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value, often zero. Solving it means finding the range of x-values where the inequality holds true. This typically requires identifying where the quadratic expression is positive or negative.

Graphing Quadratic Functions

Graphing a quadratic function y = ax² + bx + c helps visualize its shape (a parabola) and identify where it lies above or below the x-axis. The points where the graph crosses the x-axis (roots) divide the number line into intervals to test for the inequality.

Using a Graphing Utility

A graphing utility is a tool or calculator that plots functions quickly and accurately. It helps find the roots of the quadratic and shows where the graph is above or below the x-axis, making it easier to determine the solution set for the inequality.