Problem 28 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$-\frac{1}{10} x^{2}-\frac{3}{8} y<-\frac{1}{4}$$

Step-by-Step Solution

Answer

To solve the inequality, it is first rewritten as \(y > -\frac{1}{10} x^{2}-\frac{3}{8}\). Using a graphing utility, a parabolic graph is obtained. The solution to the inequality is the region above this parabola.

Step 1: Rewrite the inequality

Rewrite the inequality \(-\frac{1}{10} x^{2}-\frac{3}{8} y < -\frac{1}{4}\) so it is easier to graph. Solve for y: \(y > -\frac{1}{10} x^{2}-\frac{3}{8}\)

Step 2: Use a graphing utility to graph the inequality

Once rewritten, use a graphing utility to graph the inequality. The inequality is quadratic, resulting in a parabolic graph due to the \(x^2\) term.

Step 3: Identify the solution region

Next, identify the solution region on the graph. Because the inequality is \(y > -\frac{1}{10} x^{2}-\frac{3}{8}\), this will be the region above the parabola.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions of the second degree (typically in the form of \( ax^2 + bx + c \) where \(a, b,\) and \(c\) are constants) that are set with an inequality sign (<, >, ≤, or ≥) rather than an equals sign. For instance, the given exercise includes the inequality \( -\frac{1}{10} x^2 - \frac{3}{8} y < -\frac{1}{4} \).

To solve such inequalities, it's often useful to rearrange them into a standard form where one side is zero, such as \(y > -\frac{1}{10} x^2 - \frac{3}{8}\). The next step usually involves finding the roots of the associated quadratic equation (when the inequality is set to zero), which helps in determining the intervals where the original inequality holds true. The solutions to quadratic inequalities are often intervals or combination of intervals, and they can be graphically represented as regions on the Cartesian plane. In solving these problems, both algebraic and graphical methods are valuable, with each providing different insights into the solution.

Parabolic Graph

The graph of a quadratic function is a curve known as a parabola. These curves appear 'U' shaped and can open upwards or downwards, depending on the sign of the coefficient of the \(x^2\) term—positive opens up, negative opens down. The equation \(y = -\frac{1}{10} x^2 - \frac{3}{8}\) from the given exercise represents a downward-opening parabola because the coefficient of \(x^2\) is negative.

The vertex of the parabola is the point where the curve turns and is the maximum or minimum point of the graph. By graphing the quadratic function, students can visualize the sets of \(x\) and \(y\) that satisfy the corresponding inequality. For \(y > -\frac{1}{10} x^2 - \frac{3}{8}\), the solution region would be above the parabola since we're looking for \(y\) values greater than those on the curve. Graphing these inequalities helps in understanding the range of possible solutions and provides a visual method to validate the algebraically obtained solutions.

Graphing Utility

A graphing utility is a software tool that allows users to plot graphs, solve equations, and visualize various kinds of mathematical functions and inequalities. In the current context, a graphing utility would enable students to quickly and accurately sketch the parabola associated with the quadratic inequality \( -\frac{1}{10} x^2 - \frac{3}{8} y < -\frac{1}{4}\).

After rearranging the inequality and solving for \(y\), as seen in the step by step solution, inputting the resulting quadratic expression into a graphing utility simplifies the process of plotting. These tools often provide interactive graphs that can be manipulated for a better understanding of the problem. Moreover, they can display important features such as the vertex, axis of symmetry, and direction of opening, which are crucial in identifying the solution region for the inequality. Whether users have access to dedicated graphing calculators, computer software, or online graphing applications, these utilities are invaluable aids in studying and comprehending complex mathematical concepts like graphing inequalities.