Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{2}{x+2}-\frac{3}{x-1}$$

Step-by-Step Solution

Answer

The x-intercept of the function \(y=\frac{2}{x+2}-\frac{3}{x-1}\) is \(x=-8\).

Step 1 Title

Substitute \(y=0\) into the equation. This results in the following equation: \[0=\frac{2}{x+2}-\frac{3}{x-1}\]

Step 2 Title

Solve the equation for \(x\). To get rid of the denominator, start by finding a common denominator, which is \((x+2)(x-1)\): \[0=\frac{2(x-1)}{(x+2)(x-1)}-\frac{3(x+2)}{(x+2)(x-1)}\]Combine the fractions: \[0=\frac{2x-2-3x-6}{(x+2)(x-1)}\]Simplify the numerator:\[0=\frac{-x-8}{(x+2)(x-1)}\]For the fraction to be zero, the numerator has to be zero. Set \(-x-8=0\)

Step 3 Title

Solving \(-x-8=0\), you get \(x=-8\). This is the x-intercept.

Step 4 Title

Finally, make sure to use a graphing utility to plot the function \(y=\frac{2}{x+2}-\frac{3}{x-1}\) and confirm that it indeed intersects the x-axis at \(x=-8\).

Key Concepts

Understanding X-Intercepts

X-intercepts are the points where a graph crosses the x-axis. At these points, the value of the function is zero. In other words, for any point on the x-axis, the y-value is 0. This makes understanding x-intercepts crucial for graphing. When we're given a function and asked to find its x-intercepts, we're essentially solving the equation for when the output is zero. In the exercise, setting the function equal to zero, like this:

• Set the equation: \( y=\frac{2}{x+2}-\frac{3}{x-1} \)
• Substitute \( y = 0 \) to get \( 0 = \frac{2}{x+2}-\frac{3}{x-1} \)

By converting this to an equation, we make it possible to calculate where the graph meets the x-axis. Remember, each x-intercept corresponds to a solution of this equation.

Solving Equations Made Simple

Solving equations involves finding the value of \( x \) that makes the equation true. In rational functions, this often requires finding a common denominator. Here's a simple way to go through it:

• Identify the equation: \( 0 = \frac{2}{x+2} - \frac{3}{x-1} \)
• Find a common denominator, which is \((x+2)(x-1)\)
• Rewrite each term with this common denominator: \( \frac{2(x-1)}{(x+2)(x-1)} - \frac{3(x+2)}{(x+2)(x-1)} \)
• Combine and simplify: \( \frac{-x-8}{(x+2)(x-1)} \)

By simplifying, you can solve for \( x \) more easily. This process reveals the crucial x-intercepts, like \( x = -8 \) in our example. Remember, if the numerator solves to zero, then the entire fraction becomes zero, confirming an x-intercept.

Using a Graphing Utility

Graphing utilities can be powerful tools for visualizing how functions behave. By plotting the function \( y = \frac{2}{x+2} - \frac{3}{x-1} \), we can visually confirm our calculated x-intercepts. Here's how you might use a graphing utility:

• Enter the function into the graphing utility equation input.
• View the graph and observe where it crosses the x-axis, confirming the x-intercepts.
• Check if the plot indeed intersects the x-axis at \( x = -8 \).

Seeing the graph can reassure that the calculations are correct, and also provide insights into the behavior and shape of the rational function. It helps in understanding the overall relationship between x and y values.

Finding the Common Denominator

A common denominator is key to simplifying rational expressions. When two fractions have a different denominator, you must find a common one to combine them. Here's the step-by-step:

• Look at the different denominators in the equation: \( x+2 \) and \( x-1 \).
• Multiply these to find the common denominator: \( (x+2)(x-1) \).
• Use this common denominator to rewrite each fraction, e.g., \( \frac{2(x-1)}{(x+2)(x-1)} \) and \( \frac{3(x+2)}{(x+2)(x-1)} \).

Once you have a common denominator, combining and simplifying the fractions becomes straightforward. This allows you to focus on the numerators to solve the equation for x. Remember, combining fractions is much like combining like terms, and finding a common denominator is a crucial first step.