Problem 87 - Algebra and Trigonometry

Question

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ 2 x^{2}+5 x-3 \leq 0 $$

Step-by-Step Solution

Answer

Without the actual graph and utility results, we cannot provide a specific solution. However, the answer will describe the set of x-values for which the graph of the function \(2 x^{2}+5 x-3 = 0\) lies on or below the x-axis.

Step 1: Plot the Quadratic Equation

Firstly, plot the given quadratic equation \(2 x^{2}+5 x-3 = 0\) using a graphing utility. This curve is also called a parabola.

Step 2: Find the Roots of the Equation

The roots of the quadratic equation are the x -coordinates of the points where the parabola crosses the x-axis. On the plotted graph, identify these points.

Step 3: Determine the Solution Set for Inequality

As the inequality is given as \(2 x^{2}+5 x-3 \leq 0\), we are looking for the values of x where the function's value is less than or equal to zero. This corresponds to the sections of the x-axis for which the plot lies on or below the x-axis.

Step 4: Write Down the Solution

The solution to the inequality are the intervals of x values which are derived in the previous step. In interval notation, the solution represents the intervals where the graph lies on or below the x-axis.

Key Concepts

Graphing Utilities

Graphing utilities are digital tools designed to help visualize mathematical equations. They make it easier to understand complex functions by transforming them into interactive graphs. Using these tools, you can quickly plot the curve of a quadratic equation like \(2x^2 + 5x - 3 = 0\). This curve, known as a parabola, allows you to see where it intersects with the x-axis.

Here's how you can make the most of graphing utilities:

• Input the quadratic equation into the utility.
• Adjust the zoom to clearly view the intersections with the x-axis.
• Utilize built-in features to pinpoint the exact roots.

You'll find graphing utilities particularly useful for checking your solutions and visualizing problem areas.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They are characterized by the squared term \(x^2\). These equations typically have two solutions or roots, which can be real or complex numbers.

In our example, the equation is \(2x^2 + 5x - 3 = 0\). Here's a brief breakdown:

• \(a = 2\), which makes the parabola open upwards.
• \(b = 5\) and \(c = -3\) influence the shape and position of the parabola.

Solving a quadratic equation often involves finding where it equals zero, which means determining its roots. The roots are the values of \(x\) where the parabola intersects the x-axis.

Solution Set

The solution set of an inequality includes all the values that satisfy the inequality condition. For \(2x^2 + 5x - 3 \leq 0\), we need to find where this expression is less than or equal to zero.

Let's break it down:

• Plot the quadratic equation to identify the roots.
• Check where the parabola is located on or below the x-axis.

This area on the graph corresponds to the solution set, showing where the inequality holds true. It's crucial to interpret the graph correctly to find these x intervals.

Interval Notation

Interval notation is a concise way of representing a range of numbers. It is often used to express the solution set of inequalities.

For the inequality \(2x^2 + 5x - 3 \leq 0\), you will write down the intervals where the graph is below the x-axis. Here’s how:

• Locate the roots (where the graph crosses the x-axis) on the plot.
• Identify the sections of the x-axis that lie below or exactly at zero.
• Use bracket symbols: \([a, b]\) for inclusive and \((a, b)\) for exclusive bounds.

These intervals describe exactly where the function meets the inequality condition. In this way, interval notation becomes a powerful tool for clearly communicating solutions.