Solve each inequality in Exercises 86–91 using a graphing utility. 2x² + 5x - 3 ≤ 0

Verified step by step guidance

1. Identify the inequality to solve: \$2x^{2} + 5x - 3 \leq 0$.
2. Rewrite the inequality as an equation to find critical points: \$2x^{2} + 5x - 3 = 0$.
3. Use the quadratic formula to solve for \(x\): \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=2\), \(b=5\), and \(c=-3\).
4. Calculate the roots from the quadratic formula; these roots divide the number line into intervals to test for the inequality.
5. Use a graphing utility to plot \(y = 2x^{2} + 5x - 3\) and determine where the graph lies on or below the \(x\)-axis (i.e., where \(y \leq 0\)) to find the solution intervals.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to zero. Solving it requires finding the range of x-values where the quadratic expression satisfies the inequality, often by analyzing the parabola's position relative to the x-axis.

Graphing Quadratic Functions

Graphing a quadratic function y = ax² + bx + c produces a parabola. The graph helps visualize where the function is above or below the x-axis, which corresponds to positive or negative values of the quadratic expression, aiding in solving inequalities.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software, allow quick plotting of functions to identify roots and intervals where inequalities hold. They provide a visual method to solve quadratic inequalities by showing where the graph lies relative to the x-axis.