Problem 49 - 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 \(f(x)=0.1\) $$\frac{2}{x+2}=3$$

Step-by-Step Solution

Answer

By following the steps and using a graphing utility, it can be estimated that the solution of the equation is \(x \approx -1.33\).

Step 1: Simplify the Equation

Start by isolating \(x\). First, get rid of the fraction by multiplying both sides of the equation by \(x+2\). We get \(2 = 3(x + 2)\). Simplify this to obtain \(2 = 3x + 6\).

Step 2: Transpose to the Form \(f(x) = 0\)

This equation can be rewritten in the form \(f(x) = 0\) for graphing. To do this, subtract 2 from both sides to move it over to the right-hand side. This gives us \(3x + 6 - 2 = 0\), which simplifies to \(3x + 4 = 0\). So, the equation is now in the correct form for graphing: \(f(x) = 3x + 4\).

Step 3: Use a Graphing Utility

Using a graphing utility, plot the function \(f(x) = 3x + 4\). The solution(s) of the equation is/are the x-coordinate(s) at which the curve intersects the x-axis.

Key Concepts

Graphing Utility

A graphing utility is a valuable tool for visualizing functions and solving equations. In this exercise, it helps you find where the line represented by a function crosses the x-axis. These points of intersection are the solutions to the equation. Graphing utilities, like calculators or software, plot the equation so you can easily see where the function equals zero.
This visualization simplifies the process of solution-finding, especially for more complex equations beyond simple linear ones. Here, using such a tool makes it straightforward to identify the x-intercept of the equation transformed into the function form, ensuring that the solution is both clear and accurate.

Isolating x

To isolate x means to rearrange the terms in an equation so that x stands alone on one side. This process is key in solving equations, as it allows you to find the value(s) of x that make the equation true.
In our example, we started with a fractional equation \(\frac{2}{x+2} = 3\). The first step was to eliminate the fraction by multiplying both sides by \(x+2\). This gave \(2 = 3(x+2)\).
Further simplification involves distributing the 3 and then moving terms around to get "3x = -4", effectively isolating x.

Linear Equations

Linear equations form straight lines when graphed. They are crucial in algebra because of their simplicity and the neat, clear patterns they reveal.
The general form of a linear equation is usually \(ax + b = 0\). In this exercise, we rewrote the equation as \(3x + 4 = 0\). Solving such equations involves basic algebraic manipulations to find where the line cuts the x-axis.
This crossing point is where \(y = 0\), indicating the solution to the equation. Linear equations are simple yet powerful, forming the foundation of understanding more complex mathematical concepts.

Function Transformation

Function transformation involves changing how a function is expressed so it's easier to analyze. In our case, the transformation included converting the given fraction equation to a form suitable for graphing.
This transformation is achieved by algebraically manipulating the equation so it has the form \(f(x) = 0\). We arrived at \(f(x) = 3x + 4\).
Transforming functions often involves scaling, translating, or reflecting them, but here our focus was on rewriting the equation into a linear format for easier graph solution finding. Through transformations, we can view the same equation in different lights, aiding our understanding and approach to finding solutions.