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

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Rewrite the inequality to understand the expression clearly: \(x^{2} + 3x - 10 > 0\).

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 \(c=-10\).

Calculate the discriminant \(\Delta = b^{2} - 4ac\) to determine the nature of the roots.

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.

Express the solution set as intervals where the quadratic expression is greater than zero, based on the test results from the intervals.

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주요 개념

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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.

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