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

Verified step by step guidance

1. Rewrite the inequality as a quadratic expression: $\mathrm{2x}{x}^2+5x-3\le 0$.
2. Use a graphing utility to graph the quadratic function $f(x)\; =\; 2x^2\; +\; 5x\; -\; 3$ and identify where the graph lies on or below the x-axis (i.e., where $f(x)\; \le \; 0$).
3. Find the x-intercepts (roots) of the quadratic by solving $2x^2\; +\; 5x\; -\; 3\; =\; 0$ using the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$, where $a=2$, $b=5$, and $c=-3$.
4. Determine the intervals on the x-axis between and outside the roots, then test points in each interval to see where the inequality $2x^2\; +\; 5x\; -\; 3\; \le \; 0$ holds true.
5. Express the solution set as the interval(s) of x-values where the quadratic is less than or equal to zero, based on the graph and test points.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value, often zero. Solving it means finding the range of x-values where the inequality holds true. This typically requires analyzing the parabola formed by the quadratic function.

Graphing Quadratic Functions

Graphing a quadratic function y = ax^2 + bx + c produces a parabola. The direction (up or down) depends on the sign of 'a'. Using a graphing utility helps visualize where the parabola lies above or below the x-axis, which corresponds to the solution of inequalities.

Finding Roots and Intervals

The roots of the quadratic equation 2x^2 + 5x - 3 = 0 are the x-values where the parabola crosses the x-axis. These roots divide the number line into intervals. Testing points in these intervals determines where the inequality 2x^2 + 5x - 3 ≤ 0 holds true.