Solve each inequality in Exercises 86–91 using a graphing utility. x^2 + 3x - 10 > 0

Verified step by step guidance

1. Identify the inequality to solve: x^2 + 3x - 10 > 0. This is a quadratic inequality.
2. Rewrite the inequality in standard form if necessary. In this case, it is already in the form ax^2 + bx + c > 0, where a = 1, b = 3, and c = -10.
3. Use a graphing utility to graph the quadratic equation y = x^2 + 3x - 10. Look for the x-values where the graph of the equation is above the x-axis, as these x-values will satisfy the inequality.
4. Identify the x-values where the graph intersects the x-axis. These points are the roots of the equation and can be found using the graphing utility. These points divide the x-axis into intervals.
5. Test each interval determined by the roots to see if the values in those intervals make the inequality true. Choose a test point from each interval, substitute it into the inequality, and check if the inequality holds. The intervals that satisfy the inequality are part of the solution set.

Key Concepts

Inequalities

Inequalities express a relationship where one side is not equal to the other, often using symbols like '>', '<', '≥', or '≤'. In this case, the inequality x^2 + 3x - 10 > 0 indicates that we are looking for values of x that make the quadratic expression positive. Understanding how to manipulate and solve inequalities is crucial for finding the solution set.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. In this problem, the quadratic x^2 + 3x - 10 will be analyzed to determine where it is greater than zero, which involves finding its roots and analyzing the intervals.

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and inequalities. By graphing the quadratic function x^2 + 3x - 10, one can easily identify the regions where the function is above the x-axis, thus solving the inequality. Familiarity with using graphing utilities is essential for efficiently analyzing and interpreting the results of such problems.