Problem 94 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation graphically. $$|0.5 x+2|+|0.25 x+4|=9$$

Step-by-Step Solution

Answer

Graph the functions and find intersection points where the sum equals 9.

Step 1: Understand the Equation

We have the equation \(|0.5x+2|+|0.25x+4|=9\). This is a sum of two absolute value functions, which makes it a piecewise function. To solve it graphically, we'll plot each absolute value individually and find where their sum equals 9.

Step 2: Graph Each Absolute Value Function

The first part of the equation, \(|0.5x + 2|\), graphs a V-shape. Similarly, \(|0.25x + 4|\) is another V-shape. Plot these graphs by finding where each individual function changes direction. For \(0.5x + 2 = 0\), the vertex is at \(x = -4\), and for \(0.25x + 4 = 0\), the vertex is at \(x = -16\). Plot these vertices on the graph.

Step 3: Plot the Horizontal Line

To solve the equation graphically, draw the horizontal line \(y = 9\) on the same set of axes. This line represents the right-hand side of the equation \(|0.5x+2|+|0.25x+4|=9\).

Step 4: Identify Intersections

Look for points where the sum of the absolute value functions intersects with the horizontal line \(y = 9\). The x-coordinate of these intersection points are the solutions to the equation because that's where the sum of the functions equals 9.

Step 5: Verify Intersection Points Are Solutions

Read off the x-values from your graph where the line \(y = 9\) intersects with the sum of the absolute value functions. Substitute these x-values into the original equation to verify that they satisfy \(|0.5x+2|+|0.25x+4|=9\).

Key Concepts

Piecewise Functions

Piecewise functions are special kinds of mathematical expressions that are defined by different expressions over different intervals. This means that what the function equals depends on which part of the input domain you are using. In the context of solving graphical equations, piecewise functions often contain components like absolute value functions that change their behavior at certain points, called 'breakpoints'.

When working with piecewise functions, it's important to determine these breakpoints, as they signify where the function shifts from one expression to another. For instance, if a piecewise function is composed of absolute values, the vertex of each absolute value graph, where it switches direction, is a natural breakpoint. By graphing these segments correctly, you can visually assess where the entire function intersects any given value, as required in the problem-solving process. Understanding these characteristics helps to accurately interpret graphical data and solutions.

Absolute Value Functions

Absolute value functions are an important subset of piecewise functions. They possess a unique V-shaped graph when plotted, which results from the nature of absolute value itself, defined mathematically as \(|x| = x\) if \(x ≥ 0\) and \( -x\) if \(x < 0\). This duality ensures the function is always non-negative.

To graph an absolute value function such as \(|0.5x + 2|\), start by identifying the x-value that makes the inside of the absolute value zero—known as the vertex. In this scenario, the vertex for \(0.5x + 2 = 0\) is at \(x = -4\). The plot will form two linear pieces: one sloping upwards and the other downwards from this vertex.By examining each absolute value function separately—as step-by-step instructions suggest—you can focus on correctly capturing their changes in direction. This precision is critical when attempting to find the solution graphically, especially when these graphs are part of more complex sum functions.

Graphing Technique

Graphing multi-component functions involves an organized technique to ensure clarity. The process usually starts by plotting individual functions on the same set of axes. It is important to accurately measure and draw each component, such as the absolute value functions involved. These are graphed first individually to locate and define their specific vertices, setting a foundation for understanding how they sum together.

Next, one must plot the horizontal line that represents the equation's right-hand side—in this exercise, it's \(y = 9\). This line serves as a benchmark to compare against the sum of the plotted functions.

• Find Intersections: Check visually for points where the sum graph crosses the horizontal line.
• Interpret Results: Determine the x-coordinates of intersection points. These supply the solutions to the equation, indicating where the sum matches the equation's requirement.
• Verification: Finally, substitute these x-values back into the original equation to ensure they provide a valid solution, confirming the graphical findings were correct and matching the problem's conditions.

Graphing solutions entail this layered approach, reinforcing step-by-step comprehension in a process where precision in plotting directly ties into accurate problem-solving.