Problem 63

Question

$59-66$ . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{1 / 3}

Step-by-Step Solution

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Answer

The solution is the interval $(0, 1)$.

1Step 1: Set the Inequality

We have the inequality $x^{1/3} < x$ which will be analyzed by solving it graphically.

2Step 2: Define Functions

Define two functions based on the inequality: $f(x) = x^{1/3}$ and $g(x) = x$. We will examine where $f(x) < g(x)$.

3Step 3: Graph the Functions

Plot both functions on the same set of axes. $f(x) = x^{1/3}$ is the cube root function, which is increasing, and $g(x) = x$ is the line through the origin with a slope of 1.

4Step 4: Identify the Intersection Points

Determine where the graphs of $f(x)$ and $g(x)$ intersect. This happens when $x^{1/3} = x$. Solve this equation: $x^{1/3} = x$ which simplifies to $x^3 = x$ or $x(x^2-1)=0$. The solutions are $x=0$ and $x=\pm 1$.

5Step 5: Analyze Regions

Examine the regions defined by the intersection points: $(-\infty, 0)$, $(0, 1)$, and $(1, \infty)$. Determine where $x^{1/3} < x$. The cube root function is below the line for $x$ between $0$ and $1$.

6Step 6: Conclusion

From the analysis, the inequality $x^{1/3} < x$ holds true in the interval $(0, 1)$. As specific points are required, we round to two decimal places, although those values are already exact in this context.

Key Concepts

Cube Root FunctionIntersection PointsGraphical AnalysisAlgebraic Inequality Solutions

Cube Root Function

The cube root function, represented as $ f(x) = x^{1/3} $, is a fascinating mathematical concept.
Unlike the square root, the cube root can also take negative numbers and still return a real number.
This function represents the value that, when multiplied by itself three times, returns the original value $ x $.

Some interesting characteristics of the cube root function include:

It is an odd function, meaning that $ f(-x) = -f(x) $.
The graph has a gentle curve, rising slowly on both sides.
It passes through the origin, $(0,0)$, because the cube root of zero is zero.
As $ x $ increases or decreases, the function's rate of change decreases, giving it a flattened appearance as it moves away from the origin.

The understanding of this function's shape and behavior is critical for analyzing inequalities it may be part of.

Intersection Points

In the context of graphing inequalities such as $ x^{1/3} < x $, finding intersection points is key.
These points tell us where two functions are equal, helping us divide the number line into regions to examine further.

For the functions $ f(x) = x^{1/3} $ and $ g(x) = x $, the intersection occurs when these functions are equal, i.e., $ x^{1/3} = x $.
Solving for $ x $, we find that the roots are $ x = 0 $, $ x = 1 $, and $ x = -1 $.

These intersection points are our critical points:

$ x = -1 $ indicates where $ x^{1/3} $ equals $ -1 $.
$ x = 0 $ shows that the two functions meet right at the origin.
$ x = 1 $ is the point where their paths cross again.

Each intersection helps in inspecting the inequality in different regions of the number line.

Graphical Analysis

Graphical analysis is the technique where we plot graphs to solve inequalities visually.
This method is especially useful for understanding the behavior of different functions relative to each other.

In solving $ x^{1/3} < x $ graphically, we plot both $ f(x) = x^{1/3} $ and $ g(x) = x $ on the same set of axes.
The graph of $ f(x) = x^{1/3} $ is a curve that slowly rises or falls depending on $ x $, while $ g(x) = x $ is a straight line passing through the origin.
Our goal is to find where the cube root function lies below the line, indicating $ f(x) < g(x) $.

Once plotted:

Observe visually which sections of the graph have $ f(x) $ under $ g(x) $.
Inspect the behavior in distinct regions delineated by intersection points.
Finally, identify where the inequality $ x^{1/3} < x $ holds true.

This visual method offers an intuitive grasp of when one function is greater than or lesser than another.

Algebraic Inequality Solutions

Solving inequalities algebraically provides a precise and analytical approach.
For the inequality $ x^{1/3} < x $, the solution involves both understanding the functional behavior and employing algebraic manipulation.

Start by equating the functions to find critical intersection points, as seen in $ x^{1/3} = x $.
Upon simplification, this becomes $ x^3 = x $, leading to $ x(x^2 - 1) = 0 $, and hence the solutions $ x = 0 $, $ x = 1 $, and $ x = -1 $.

Next, analyze regions on the number line resulting from these intersection points:

For $(-\infty, 0)$, $ x^{1/3} > x $.
In $(0, 1)$, $ x^{1/3} < x $, confirming the inequality holds true.
And for $(1, \infty)$, $ x^{1/3} > x $ once more.

Thus, via algebraic solutions, we conclude the inequality is valid solely within the interval $ (0,1) $.
These solutions are exact, even if approximations to two decimal places are conventionally stated for aesthetic reasons.

Problem 63

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