Problem 82 - 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. $$y=x-1-\frac{2}{2 x-3}$$

Step-by-Step Solution

Answer

The \(x\)-intercepts are the solutions to the equation, obtained after graphing and solving.

Step 1: Graphing the Function

To graph the function, a graphing utility is used since it simplifies the process. The function \(y = x-1 - \frac{2}{2x-3}\) is put in and the path of the line is observed.

Step 2: Finding x-intercepts from the Graph

From the graph, observe where the line crosses the \(x\)-axis. These are the \(x\)-intercepts.

Step 3: Solving the Equation

To confirm the \(x\)-intercepts, set \(y\) equal to zero and solve the equation. Hence, \(x-1 - \frac{2}{2x-3} = 0\). Solve for \(x\) here.

Step 4: Checking the Solution

Check the solution(s) by substituting back into the original equation to ensure it equates to zero. If true, they are valid \(x\)-intercepts.

Key Concepts

Exploring x-intercepts

Understanding x-intercepts can be both fun and important for graphing functions. Essentially, an x-intercept is where a graph touches or crosses the x-axis. At this point, the y-value is zero. So, we can find the x-intercepts by setting the equation of the function equal to zero and solving for x. For example, with our function \( y = x - 1 - \frac{2}{2x-3} \), we set \( y = 0 \) and solve:

• Set the function equal to zero: \( x - 1 - \frac{2}{2x-3} = 0 \).
• Solve for \( x \) to find where the graph crosses the x-axis.

Using a graphing utility helps visualize the exact point, giving a clearer picture of where these intercepts occur.

Solving Equations to Find Intercepts

Solving the equation is a vital step in confirming the x-intercepts. This process involves algebraic manipulation to find value(s) of \( x \) that satisfy the equation when substituted back. With our example, we have the equation:

• Start by isolating the fractional term: \( x - 1 - \frac{2}{2x-3} = 0 \).
• Add \( \frac{2}{2x-3} \) to both sides: \( x - 1 = \frac{2}{2x-3} \).
• Multiply both sides by \( 2x - 3 \) to clear the fraction.
• Simplify and rearrange the equation to find the solution for \( x \). This will help us determine the points where the x-intercepts occur.

This method is powerful because it assures precision, especially when confirmed graphically.

Applying the Substitution Method

The substitution method is useful for verifying any hypothetical solutions we find. After solving the equation for the x-intercepts, substitute these values back into the original equation. Here's a simple breakdown:

• First, identify the solution found, say \( x = a \).
• Substitute \( x = a \) back into the original function \( y = x - 1 - \frac{2}{2x-3} \).
• If your substitution results in \( y = 0 \), then \( x = a \) is indeed an x-intercept.

This substitution process is crucial because it lets you confirm that your algebraic manipulations were correct and your intercepts are exactly as calculated. Thus, this method solidifies your findings with certainty.