Problem 71 - A Graphical Approach to College Algebra

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) \( x \in (-\infty, -3.54) \cup (1.32, \infty) \); (c) \( x \in (-3.54, 1.32) \).

Step 1: Understand the equation

We start with the equation \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}=0 \). This means that the numerator needs to be zero for the fraction to equal zero, as a non-zero denominator avoids division by zero.

Step 2: Solve the numerator for zero

Set the numerator equal to zero: \( \sqrt{2} x + 5 = 0 \). Solve for \(x\) by subtracting 5 from both sides: \( \sqrt{2} x = -5 \), then divide both sides by \(\sqrt{2}\): \( x = \frac{-5}{\sqrt{2}} \). Simplify to get \( x \approx -3.54 \) when rounded to the nearest hundredth.

Step 3: Determine domain restrictions

The denominator \( x^3 - \sqrt{3} \) must not be zero, as division by zero is undefined. Solve \( x^3 - \sqrt{3} = 0 \) to find points where the function is undefined: \( x = \sqrt[3]{\sqrt{3}} \). Calculate this to be approximately \( x \approx 1.32 \).

Step 4: Graph the function

Graph the function \( y = \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} \) using a graphing calculator or software. Identify where the graph crosses the x-axis at \( x \approx -3.54 \). Notice vertical asymptotes at \( x \approx 1.32 \).

Step 5: Solve Inequality (b) Graphically

For \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}>0 \), observe the graph to see where it lies above the x-axis. Note that the function is above the x-axis at intervals \( (-\infty, -3.54) \cup (1.32, \infty) \).

Step 6: Solve Inequality (c) Graphically

For \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}<0 \), observe the graph where it lies below the x-axis. The function is below the x-axis at interval \( (-3.54, 1.32) \).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values, where these values are not necessarily equal. In simple terms, they tell us one amount is bigger or smaller than the other.
When dealing with inequalities like the ones in the exercise, the goal is to find the range of values of the variable that make the inequality true.
In this example:

• The inequality \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}>0 \) asks us to find values of \( x \) where the function is greater than zero. In simpler words, where is the function positive or above the x-axis on the graph?
• Conversely, the inequality \( \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}<0 \) seeks the values of \( x \) for which the function is less than zero, meaning it's negative or below the x-axis.

Solving inequalities graphically often makes things clearer, as visualizing where a curve is above or below the x-axis helps you identify these ranges easily. A graph provides a straightforward picture of where the expression shifts from positive to negative or vice versa.

Diving into Domain Restrictions

Domain restrictions are crucial when dealing with functions, especially when fractions or square roots are involved in the expression. The domain refers to the set of all possible values that a variable, typically \( x \), can take for which the function remains valid.
In terms of domain restrictions, it helps prevent mathematical impossibilities like division by zero. For the equation in question, the denominator is \( x^3 - \sqrt{3} \), which should not be zero. Solving \( x^3 - \sqrt{3} = 0 \) gives us a domain restriction, finding \( x = \sqrt[3]{\sqrt{3}} \), or approximately \( x \approx 1.32 \).
When graphing or solving inequalities, these restrictions translate to vertical asymptotes in the graph of the function. That means the function's curve will approach a certain value (\( x \approximately 1.32\) for this problem) without ever touching or crossing it. Recognizing and factoring in these restrictions ensures you accurately address the behavior of the function across its valid range.

Essentials of Function Graphing

Graphing a function involves plotting it on a coordinate plane to visualize its behavior. This step is instrumental in understanding the complete dynamics of complex functions like fractions or those involving roots.
For the function \( y = \frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}} \), start by graphing it carefully, ensuring all elements are considered:

• Points where the function crosses the x-axis (the numerator becomes zero) highlight where the function equals zero, specifically at \( x \approx -3.54 \).
• Look for vertical asymptotes caused by domain restrictions like \( x \approx 1.32 \), where the function is undefined.

By graphing, you can clearly identify regions where the function is positive or negative, aiding in solving inequalities. Employing technology, such as graphing calculators or apps, can significantly enhance accuracy and convenience, giving you a precise graphical interpretation that supports analytical solutions.