Problem 55 - 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 \(f(x)=0.1\) $$|3 x-2|-1=4$$

Step-by-Step Solution

Answer

The solutions to the equation \(|3x-2|-1=4\) are \(x = 7/3\) and \(x = 1\).

Step 1: Rewrite the equation

Let's rewrite the equation to the form \(f(x)=0\). This means that \(|3x-2|-1-4=0\), which simplifies to \(|3x-2|-5=0\). So we have our function \(f(x) = |3x-2|-5\).

Step 2: Deal with the absolute value

An equation with absolute value \(|p|=q\) is equivalent to \(p=q\) or \(p=-q\). Applying this rule, we get two equations: \(3x-2-5=0\) or \(-(3x-2-5)=0\). Simplifying these leads to \(3x=7\) or \(3x=3\).

Step 3: Solve the equations

Now we solve the two equations for x. The solutions are \(x = 7/3\) and \(x = 1\).

Step 4: Use a graphing utility

Use a graphing utility to plot the function \(f(x) = |3x-2|-5\). The roots of the equation are the x-coordinates of the points where the plot crosses the x-axis. These should correspond to the solutions we found analytically in Step 3.

Key Concepts

Graphing Utility

A graphing utility is an indispensable tool when dealing with complex equations, mainly because it provides a visual representation. For the equation \(|3x-2|-1=4\), expressing it in the standard form of \(f(x) = 0\) simplifies the plotting process. This involves rewriting the equation as \(|3x-2|-5=0\).

• Key feature: By graphing \(f(x) = |3x-2|-5\), you can visualize the part of the graph that intersects the x-axis. These points of intersection represent the solutions to the equation.
• Advantages: Graphing utilities allow you to see the solution in real-time, which helps in understanding the nature and behavior of the equation.
• How to use: Input the function into the graphing utility, and it produces a graph with the x-axis crossings displaying the solutions clearly.

This method complements hand-solving techniques by offering quick accuracy checks and giving you a graphical insight into the solutions you've calculated.

Solving Linear Equations

Linear equations form the backbone of algebra, making it crucial to understand how they're solved. After transforming our absolute value equation into \(|3x-2|-5=0\), we have two linear equations: \(3x=7\) and \(3x=3\).
Here's how you can solve linear equations effortlessly:

• Isolate the variable: Starting with \(3x=7\), divide both sides by 3 to get \(x = \frac{7}{3}\).
• Repeat for the second equation: Similarly, from \(3x=3\), divide by 3 to obtain \(x = 1\).
• Check solutions: Substitute the solutions back into the original equation to ensure they satisfy it.

The solutions \(x = \frac{7}{3}\) and \(x = 1\) are the points where the linear representations intersect the x-axis in the graph of \(f(x) = |3x-2|-5\). This confirms the accuracy of both algebraic and graphical methods.

Piecewise Functions

In dealing with absolute value equations, understanding piecewise functions is essential. The equation \(|3x-2|\) can be expressed as two linear equations, depending on the value of \(x\).

• If \(3x-2\geq 0\), meaning \(x \geq \frac{2}{3}\), the expression is simply \(3x-2\).
• If \(3x-2 < 0\), meaning \(x < \frac{2}{3}\), the expression becomes \(-(3x-2)\).

This leads us to express the function piecewise as:
\[ f(x) = \begin{cases} 3x-2-5, & \text{if } x \geq \frac{2}{3} \ -(3x-2)-5, & \text{if } x < \frac{2}{3} \end{cases} \]
Such a format is extremely useful when graphing, as it breaks down the function into manageable pieces that behave linearly within their specific domains.

Graphical Solution Methods

Graphical solution methods bring a conceptual understanding by visualizing the solutions of equations. By plotting the function \(f(x) = |3x-2|-5\), the points where it hits the x-axis show the solutions \(x = \frac{7}{3}\) and \(x = 1\).

• Graph interpretation: The crossings of the x-axis represent zero points, providing direct visual confirmation of solutions calculated analytically.
• Advantages: It's a powerful way to intuitively check the validity of solutions and visually detect additional characteristics of the function, such as symmetry and intercepts.
• Application: For more complex scenarios, the graph helps to see if there are other additional solutions or if there's behavior the algebraic approach might miss.

This method serves both as a tool for checking solutions and understanding the broader characteristics of the function, making it a vital aspect of solving equations involving absolute values.