Problem 53 - Algebra and Trigonometry

Question

Find the solutions of the inequality by drawing appropriate graphs. State each answer correct to two decimals. $$ x^{1 / 3}

Step-by-Step Solution

Answer

The solutions are \((-1, 0) \cup (1, \infty)\).

Step 1: Understand the inequality

The inequality is \( x^{1/3} < x \). This means we are comparing the cube root of \( x \) (\( x^{1/3} \)) with \( x \) itself. We need to find the values of \( x \) for which this inequality holds true.

Step 2: Identify the functions

We have two functions to compare: \( f(x) = x^{1/3} \) and \( g(x) = x \). We need to plot both functions on a graph to determine where \( f(x) \) is less than \( g(x) \).

Step 3: Graph the functions

Graph \( f(x) = x^{1/3} \) which is the cube root of \( x \), and it is typically a curve that grows more slowly than the linear function \( g(x) = x \). Graph \( g(x) = x \) which is a straight line passing through the origin with a slope of 1.

Step 4: Analyze the graph intersection points

Find the point where the two graphs intersect. This is where \( x^{1/3} = x \). For this, solve the equation \( x^{1/3} = x \). Raise both sides to the power of 3 to get \( x = x^3 \). Simplify this to \( x^3 - x = 0 \). Factor this equation to \( x(x^2 - 1) = 0 \), which gives solutions \( x = 0, x = 1, x = -1 \).

Step 5: Determine the intervals

To find the intervals where \( x^{1/3} < x \), check the intervals determined by the intersection points: \((-\infty, -1)\), \((-1, 0)\), \((0, 1)\), and \((1, \infty)\). Evaluate a test point in each interval to check where the inequality holds.

Step 6: Test interval lengths separately

Choose \( x = -2 \) in \((-\infty, -1)\), giving \((-2)^{1/3} \approx -1.26\) and \( -2 \), so \((-2)^{1/3} > -2\). Choose \( x = -0.5 \) in \((-1, 0)\), \((-0.5)^{1/3} \approx -0.7937\), so \((-0.5)^{1/3} < -0.5\). Choose \( x = 0.5 \) in \((0, 1)\), giving \(0.5^{1/3} \approx 0.7937\), so \(0.5^{1/3} > 0.5\). Choose \( x = 2 \) in \((1, \infty)\), then \(2^{1/3} \approx 1.26\), so \(2^{1/3} < 2\).

Step 7: Compile the solution

From the test intervals, the inequality \( x^{1/3} < x \) holds in the intervals \((-1, 0)\) and \((1, \infty)\). Therefore, the solutions to the inequality are \((-1, 0) \cup (1, \infty)\). Include each point of the solution up to two decimal places as required.

Key Concepts

Graphical Method

To effectively solve inequalities like \( x^{1/3} < x \), a graphical method can be a powerful tool. By plotting the functions on a coordinate plane, we can visually inspect where one function falls below the other. This method simplifies the process by showing a clear visual representation of where one function is less than the other.

Here, we are dealing with two functions: \( f(x) = x^{1/3} \) and \( g(x) = x \). The graphical method involves drawing both of these on the same set of axes. The function \( f(x) = x^{1/3} \) will appear as a cube root curve, which means it grows slower compared to the other linear function, \( g(x) = x \). By tracing each function's path, intersections become apparent.

With this approach, you will see a curve below a line, showing intervals where the inequality holds. These intersections serve as crucial points dividing the graph into segments, where testing each segment ensures whether the inequality is true.

Cube Root Functions

Cube root functions like \( f(x) = x^{1/3} \) have a distinctive shape and behavior, characterized by their slower growth compared to linear functions. They are symmetric around the origin and grow at a decreasing rate, which can be visualized as a smooth S-shaped curve passing through the origin.

When plotted, the cube root function will have the general characteristics of slowly rising and dipping to either side. One primary feature of cube root functions is their slower increase compared to their linear counterparts, making them helpful for identifying inequalities where \( x^{1/3} < x \).

In calculus or algebra, understanding the characteristics of cube root functions enables better prediction of solutions to inequalities. By grasping these characteristic behaviors, assessing their comparison with other functions becomes simpler.

Interval Testing

Interval testing is an essential approach to confirming solutions to inequalities after identifying intersection points graphically. By checking various test points within each interval, we can determine in which parts of the graph the inequality \( x^{1/3} < x \) holds.

Here’s how it works:

• Select intervals based on the intersection points, such as \((-\infty, -1), (-1, 0), (0, 1), \text{and} (1, \infty)\).
• Choose any test point within each interval, and substitute back into the inequality \( f(x) = x^{1/3} < g(x) = x \).
• If the inequality holds true for the test point, it is true for that entire interval.

This method ensures accuracy by sampling and confirming the inequality's truth across designated zones, using test values like \( x = -2 \), \( x = -0.5 \), \( x = 0.5 \), and \( x = 2 \).

Function Intersection Points

Function intersection points are critical for solving inequalities graphically. They represent the values of \( x \) where \( x^{1/3} = x \), marking the boundaries between segments where the inequality could change.

To find these points analytically, set the functions equal: \( x^{1/3} = x \). By manipulating this equation to \( x^3 = x \), and simplifying to \( x(x^2 - 1) = 0 \), we find \( x = 0, 1, -1 \).

These intersection points provide us with critical intervals to test. They indicate zones that make checking where \( x^{1/3} < x \) holds more manageable and precise, as these points directly influence intervals to test. Recognizing intersections helps split the solution process into manageable assessments of whether the inequality remains valid before and after each point.