Problem 43 - A Graphical Approach to College Algebra

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[4]{3 x+1}=1\) (b) \(\sqrt[4]{3 x+1}>1\) (c) \(\sqrt[4]{3 x+1}<1\)

Step-by-Step Solution

Answer

(a) \(x = 0\); (b) \(x > 0\); (c) \(x < 0\).

Step 1: Solve the Equation Analytically

To solve the equation \( \sqrt[4]{3x+1} = 1 \), first eliminate the fourth root by raising both sides of the equation to the power of 4. This gives us \( 3x + 1 = 1^4 \), simplifying to \( 3x + 1 = 1 \). Subtract 1 from both sides to get \( 3x = 0 \). Finally, divide both sides by 3 to isolate \( x \): \( x = 0 \).

Step 2: Graph the Function

Plot the function \( y = \sqrt[4]{3x+1} \) versus \( y = 1 \) on the xy-plane. The function is a fourth root function, typically increasing and smooth, plotted in the context of the fourth root of the linear function \( 3x + 1 \). The line \( y = 1 \) is a horizontal line that intersects the fourth root function at the calculated solution from part (a) at \( x = 0 \).

Step 3: Solve the Inequality \(\sqrt[4]{3x+1} > 1\) Using the Graph

From the graph, identify the regions where the function \( \sqrt[4]{3x+1} \) is above the line \( y = 1 \). Since the function is equal to 1 at \( x = 0 \) and is increasing, it will be greater than 1 when \( x > 0 \). Therefore, the solution is \( x > 0 \).

Step 4: Solve the Inequality \(\sqrt[4]{3x+1} < 1\) Using the Graph

From the graph, observe where the function \( \sqrt[4]{3x+1} \) is below the line \( y = 1 \). The function decreases to values less than 1 as \( x \) goes into negative values. Thus, the solution to the inequality is \( x < 0 \).

Key Concepts

Solving Equations

When solving algebraic equations, the goal is to find the value of the variable that makes the equation true. For the equation \( \sqrt[4]{3x+1} = 1 \), the first step is to understand the structure of the equation, which includes a fourth root. To eliminate this fourth root, raise both sides of the equation to the power of 4. This will help simplify the equation into a more manageable form:

• Starting with \( \sqrt[4]{3x+1} = 1 \), raise both sides by 4, converting it to \( 3x + 1 = 1^4 \).
• Simplifying this equation results in the linear equation \( 3x + 1 = 1 \).
• Subtract 1 from both sides, yielding \( 3x = 0 \).
• Finally, divide by 3 to solve for \( x \): \( x = 0 \).

Remember, solving algebraic equations often involves reversing operations to isolate the variable.

Graphical Representation

Graphical representation is a powerful tool in algebra. It provides a visual understanding of the solution and the behavior of an equation. In this exercise, we plot the function \( y = \sqrt[4]{3x+1} \) on the xy-plane. This function represents the fourth root of the expression \( 3x+1 \) and is plotted against the line \( y = 1 \).
The plot of \( y = \sqrt[4]{3x+1} \) is characterized by:

• Being a smooth increasing curve due to the properties of the fourth root function.
• Intersecting the line \( y = 1 \) at the point where \( x = 0 \), which is the solution found analytically.

This graphical method not only confirms the equation's solution but also helps visualize the scenarios of the inequalities \( \sqrt[4]{3x+1} > 1 \) and \( \sqrt[4]{3x+1} < 1 \). The graph makes it easier to see where the function is greater than or less than the horizontal line \( y = 1 \).

Inequalities in Algebra

Inequalities in algebra present scenarios where we determine not exact values but ranges that satisfy a condition. Based on the graphical representation:
- For \( \sqrt[4]{3x+1} > 1 \), we determine where the fourth root function is above the line \( y = 1 \). Observations from the graph show that this occurs for \( x > 0 \). The inequality describes a range, indicating that any positive value of \( x \) satisfies the inequality.
- Conversely, for \( \sqrt[4]{3x+1} < 1 \), we find the function below \( y = 1 \). This scenario happens for \( x < 0 \), representing all negative values of \( x \).
When addressing inequalities:

• Always consider the direction of the inequality (greater than or less than) based on the graph.
• Be mindful that the solution describes a range of possible values rather than just one specific answer.

Understanding how to interpret these visual cues is crucial to solving and verifying inequalities.