Problem 44 - 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) \(\sqrt[4]{x-15}=2\) (b) \(\sqrt[4]{x-15}>2\) (c) \(\sqrt[4]{x-15}<2\)

Step-by-Step Solution

Answer

(a) \(x = 31\); (b) \(x > 31\); (c) \(15 \leq x < 31\).

Step 1: Solve the Equation

To solve the equation \( \sqrt[4]{x-15} = 2 \), we first eliminate the fourth root by raising both sides to the power of 4. This gives us \( x - 15 = 2^4 \), which simplifies to \( x - 15 = 16 \). Adding 15 to both sides, we find \( x = 31 \).

Step 2: Graph the Function

Next, we graph the function \( y = \sqrt[4]{x-15} \). Plot the point \((31, 2)\), since we know \( \sqrt[4]{31-15} = 2 \). The fourth root function starts rising steeply after \( x = 15 \) and is defined for \( x \geq 15 \). The horizontal line \( y = 2 \) will intersect the curve at this point.

Step 3: Solve the Inequality \( \sqrt[4]{x-15} > 2 \)

Using the graph, identify where the curve \( \sqrt[4]{x-15} \) is above the line \( y = 2 \). Since the curve lies above this line for \( x > 31 \), the solution for \( \sqrt[4]{x-15} > 2 \) is \( x > 31 \).

Step 4: Solve the Inequality \( \sqrt[4]{x-15} < 2 \)

Look at the graph for where the curve is below the line \( y = 2 \). The curve is below this line for values \( 15 \leq x < 31 \), since the function starts from 0 at \( x = 15 \) and approaches but does not reach \( y = 2 \) as \( x \) approaches 31. Thus, the solution is \( 15 \leq x < 31 \).

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics that involves finding the value(s) of a variable that make an equation true. In this particular exercise, we solve the equation \( \sqrt[4]{x-15} = 2 \). To eliminate the fourth root, we raise both sides of the equation to the fourth power. This operation allows us to simplify the equation to a linear form: \( x - 15 = 16 \). Adding 15 to both sides results in \( x = 31 \). The solution \( x = 31 \) represents the value of \( x \) that satisfies the original equation when plugged back in. This stepwise approach is typical when handling equations that involve roots or exponents. Ensuring that each operation is correctly applied is crucial for arriving at the correct solution.

Graphical Solutions

Graphical solutions offer a visual way to understand mathematical problems. After finding the equation's solution, plotting the function \( y = \sqrt[4]{x-15} \) helps to confirm the answer. The graph of this function begins at \( x = 15 \) and rises steeply. It intersects the horizontal line \( y = 2 \) at the point \( (31, 2) \), confirming the solution derived algebraically. Graphing is not only a means of verifying solutions but also provides insight into the behavior of functions. It helps to

• Visualize where the function is defined and undefined
• Identify the nature of the function's change over the interval
• Describe interactions between different functions or expressions

Working with graphs supplements analytical methods by bringing abstract concepts into a more comprehensible form.

Inequalities

Inequalities involve finding the range of values that satisfy a given condition. In this exercise, with the graphs in hand, we now solve two inequalities: \( \sqrt[4]{x-15} > 2 \) and \( \sqrt[4]{x-15} < 2 \). Using graphical analysis, we determine where the function \( \sqrt[4]{x-15} \) lies relative to the line \( y = 2 \).

• To solve \( \sqrt[4]{x-15} > 2 \), note that for \( x > 31 \), the graph of the function is above the line \( y = 2 \), indicating the solution is \( x > 31 \).
• For \( \sqrt[4]{x-15} < 2 \), the graph is below the line from \( x = 15 \) up to \( x = 31 \), hence, the solution is \( 15 \leq x < 31 \).

These solutions are confirmed visually, showcasing the effectiveness of combining graphical insights with analytical methods to solve inequalities. This approach provides a clear and comprehensive understanding of how the functions interact with the chosen inequality conditions.