Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\frac{1}{x-3}=-2$$

Step-by-Step Solution

Answer

The approximate solution to the equation \(\frac{1}{x-3}=-2\) is x approximately equals to 3.5

Step 1: Rearrange equation

Start by rearranging the equation to look like \(f(x)=0\). This can be achieved by applying basic algebraic manipulation so that on one side of the equation we get the function \(f(x)\), and on the other side we get 0. The equation \(\frac{1}{x-3}=-2\) is rearranged in the following way: \(\frac{1}{x-3} +2=0\)

Step 2: Plot graph

Then, plot the function \(f(x)= \frac{1}{x-3} + 2\) on a graphing utility of your choice.

Step 3: Find solutions

The x-values at which the function equals zero (where the graph intersects the x-axis) are the solutions to the equation. Observe the plotted function and approximate the x-values for these points.

Key Concepts

Graphing Utility

A graphing utility is an indispensable tool in modern mathematics education, offering a dynamic approach to understanding equations and their solutions. When solving rational equations like \(\frac{1}{x-3}=-2\), graphing utilities can turn abstract algebraic expressions into visual representations, making it easier to identify solutions.

Programs such as graphing calculators, computer algebra systems (CAS), and various online graphing tools allow students to quickly plot functions and analyze their behavior. By inputting the function derived from the equation \(\frac{1}{x-3} + 2 = 0\), students can observe the curve and find the x-value(s) where the function crosses the x-axis. These intersections represent the solutions to the equation.

Using a graphing utility also aids in understanding the properties of the function, such as asymptotes, domain, and range. For instance, the rational function in the given problem has a vertical asymptote at \(x=3\), which is evident when viewed on the graph. This visual insight adds depth to a student's comprehension, transcending beyond rote algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation encompasses the techniques used to rearrange equations and expressions to facilitate their solving or simplification. For the given equation \(\frac{1}{x-3} = -2\), transforming it into the form \(f(x) = 0\) is the initial essential step for both algebraic solutions and graphing.

Some key processes include combining like terms, factoring, expanding polynomials, and dealing with fractions. With the example \(\frac{1}{x-3} + 2 = 0\), we added 2 to both sides of the initial equation to isolate the term with \(x\). This action simplifies the equation and prepares it for further analysis or graphing.

Developing proficiency in algebraic manipulation is crucial as it builds the foundation for higher-level mathematics, including calculus and differential equations. Practicing these skills reinforces a student's problem-solving capabilities, allowing for tackling complex and diverse mathematical challenges.

Function Graphing

Function graphing is a powerful technique unmatched in its ability to provide insights into the behavior of mathematical models. It illustrates how a function behaves over an interval and where it intersects the axes, revealing crucial features like roots, intercepts, intervals of increase and decrease, concavity, and more.

For our rational equation, plotting \(f(x) = \frac{1}{x-3} + 2\) on the Cartesian plane provides immediate visual feedback. The graphical representation makes it apparent that the function has no intercepts on the y-axis and that finding where the curve crosses the x-axis offers the equation's solution(s).

Understanding the nuances of function graphing allows students to effectively interpret mathematical phenomena and anticipate the results of varying different parameters. It's an essential skill for science, technology, engineering, and mathematics (STEM) fields, where graph models are prevalent and can convey vast amounts of information at a glance.