Problem 58 - College Algebra with Modeling and Visualization

Question

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{1}{x^{2}}>0 $$

Step-by-Step Solution

Answer

The inequality holds for all real numbers except \(x = 0\).

Step 1: Understand the Inequality

The given inequality is \(\frac{1}{x^{2}} > 0\). The goal is to determine for which values of \(x\) the expression \(\frac{1}{x^{2}}\) is greater than zero.

Step 2: Analyze the Denominator

The denominator \(x^2\) is the square of \(x\). A square of any real number (other than zero) is always positive. Therefore, \(x^2 > 0\) for all \(x eq 0\).

Step 3: Solve the Inequality Symbolically

Since \(\frac{1}{x^{2}}\) has its domain where \(x^2 eq 0\), it must be \(x eq 0\). Given that \(x^2 > 0\) for all \(x eq 0\), the inequality \(\frac{1}{x^{2}} > 0\) holds true for all \(x eq 0\).

Step 4: Graphically Solve the Inequality

Plot the function \(y = \frac{1}{x^2}\). Notice that the graph approaches the x-axis but never touches or crosses it, indicating that \(\frac{1}{x^2}\) remains positive for all defined values of \(x\) except at \(x = 0\), where it is undefined.

Key Concepts

Symbolic Solving

In order to solve a rational inequality symbolically, you need to break down the inequality into understandable parts. Take, for instance, the inequality \( \frac{1}{x^2} > 0 \). Here, your task is to find out which values of \( x \) satisfy the condition that \( \frac{1}{x^2} \) is greater than zero.
### Steps for Symbolic Solving:

• **Analyze the inequality:** Begin by understanding the structure of the inequality. Notice that dividing by \( x^2 \) is the same as multiplying by \( \frac{1}{x^2} \).
• **Examine the denominator:** The term \( x^2 \) represents a squared number. This is significant because any number squared, except zero, is always positive. This means \( x^2 > 0 \) for all \( x eq 0 \).
• **Solve the inequality:** Since the denominator \( x^2 \) is never zero (except when \( x \) is zero), the entire expression \( \frac{1}{x^2} \) is positive for all \( x eq 0 \). Hence, \( \frac{1}{x^2} > 0 \) is true everywhere except \( x = 0 \), where it is undefined.

Thus, using symbolic solving helps you see that any \( x \) except zero keeps the expression positive.

Graphical Representation

Graphical representation gives a visual understanding of rational inequalities. Let's consider the function \( y = \frac{1}{x^2} \) and how it behaves when plotted.
### Visualizing Graphically:

• **Graph the function:** Start by sketching the graph of \( y = \frac{1}{x^2} \). Notice that as \( x \) moves away from zero either positively or negatively, the function goes upwards, forming a U-shaped curve.
• **Observe the graph at zero:** The function is undefined at \( x = 0 \) because you cannot divide by zero. But as \( x \) gets closer to zero, the values of \( y \) get extremely large.
• **Analyze the x-axis:** The graph never touches the x-axis. This indicates that \( \frac{1}{x^2} \) never becomes zero or negative; it remains positive for all other values of \( x \).

This graphical representation confirms the symbolic analysis, providing more insight through visual interpretation.

Denominator Analysis

Analyzing the denominator in rational inequalities is crucial since it directly influences whether an expression is defined and whether it can change sign.
### Understanding Denominator Analysis:

• **Importance of the denominator:** In a fraction like \( \frac{1}{x^2} \), the denominator \( x^2 \) is enforced by its attributes. Because \( x^2 \) is a squared number, it's inherently non-negative (\( x^2 \geq 0 \)).
• **Behavior of squared expressions:** For any real number other than zero, \( x^2 \) remains strictly positive. This means that within the domain where \( x eq 0 \), \( x^2 \) ensures the entire fraction stays positive.
• **Key takeaway:** Analyzing the denominator helps conclude that the expression remains undefined when \( x = 0 \) and positive elsewhere. It's a fundamental part of determining where the inequality holds true across real numbers.

By examining the denominator, you establish a vital component of solving rational inequalities successfully.