Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Solve the equation in part (a) graphically, expressing solutions to the nearest hundredth. Then use the graph to solve the associated inequalities in parts (b) and (c), expressing endpoints to the nearest hundredth. (a) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}=0\) (b) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}>0\) (c) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}<0\)

Step-by-Step Solution

Answer

(a) \( x \approx -3.54 \); (b) \( x < -3.54 \) or \( x > \sqrt[3]{\sqrt{3}} \); (c) \(-3.54 < x < \sqrt[3]{\sqrt{3}} \).

Step 1: Identify Critical Points

To solve the equation graphically, first identify critical points by setting the numerator and denominator equal to zero separately. The numerator \( \sqrt{2}x + 5 = 0 \) gives the critical point \( x = -\frac{5}{\sqrt{2}} \). The denominator \( x^3 - \sqrt{3} = 0 \) gives the critical point \( x = \sqrt[3]{\sqrt{3}} \).

Step 2: Graph the Function

Graph the function \( f(x) = \frac{\sqrt{2}x + 5}{x^3 - \sqrt{3}} \) using these critical points. Plot the vertical asymptote at \( x = \sqrt[3]{\sqrt{3}} \) and mark the x-intercept at \( x = -\frac{5}{\sqrt{2}} \).

Step 3: Solve Part (a)

To find where the function is zero, locate the x-intercept on the graph. The solution to \( \frac{\sqrt{2}x + 5}{x^3 - \sqrt{3}} = 0 \) is the x-value of the x-intercept, which is \( x = -\frac{5}{\sqrt{2}} \). This value approximates to \( x \approx -3.54 \).

Step 4: Solve Part (b)

For \( \frac{\sqrt{2}x + 5}{x^3 - \sqrt{3}} > 0 \), look at the graph and determine where the function is above the x-axis. In this case, the function is positive when \( x < -3.54 \) and \( x > \sqrt[3]{\sqrt{3}} \).

Step 5: Solve Part (c)

For \( \frac{\sqrt{2}x + 5}{x^3 - \sqrt{3}} < 0 \), identify where the function is below the x-axis. The function is negative for \( -3.54 < x < \sqrt[3]{\sqrt{3}} \).

Key Concepts

Critical Points

To begin graphing rational functions like \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} \), it's important to identify critical points. These are points where the graph can change direction or has significant features like asymptotes or intercepts.

First, set the numerator equal to zero:

• \( \sqrt{2}x + 5 = 0 \) gives \( x = -\frac{5}{\sqrt{2}} \), which is an x-intercept.

Next, set the denominator equal to zero:

• \( x^3 - \sqrt{3} = 0 \) gives \( x = \sqrt[3]{\sqrt{3}} \), indicating a vertical asymptote.

These critical points lay the foundation for understanding the behavior of the graph and are integral in solving both the equation and related inequalities. By pinpointing the numerators and denominators zero, we determine where the function is undefined or crosses the x-axis.

Vertical Asymptotes

Vertical asymptotes represent places where a function's value becomes undefined, often where the denominator of a rational function equals zero. They are crucial in sketching the graph.

For the equation \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} \), the vertical asymptote is located at \( x = \sqrt[3]{\sqrt{3}} \). This value happens because the denominator becomes zero, making the function undefined at that point.

On a graph, this is represented by a dashed vertical line at \( x = \sqrt[3]{\sqrt{3}} \), showing that the curve approaches this line but never actually touches it. Asymptotes provide important boundaries for the function's behavior and are evident when solving inequalities as well, since they define intervals where values transition from positive to negative, or vice versa.

Solving Inequalities Graphically

Graphical solutions can be very insightful, particularly with inequalities. To solve inequalities in this exercise, we graph the function \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} \).

1. **For \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} > 0 \):**
Look where the graph is above the x-axis. The solution occurs when \( x < -3.54 \) and \( x > \sqrt[3]{\sqrt{3}} \), showing regions where the numerator's influence outweighs that of the denominator, making the function positive.
2. **For \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} < 0 \):**
Locate where the graph is below the x-axis. This position appears for \( -3.54 < x < \sqrt[3]{\sqrt{3}} \), indicating intervals where the numerator results in the function being negative.

Graphically solving inequalities gives immediate visualization of where a function holds particular values, simplifying the problem of determining positive and negative intervals in relation to the x-axis.