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Chapter 4: Problem 61

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{5}{x^{2}-4}<0 $$

Short Answer

Expert verified
The solution is \((-2, 2)\).

Step by step solution

01

Determine the Domain

Identify where the expression \( x^2 - 4 \) is defined. Since we cannot divide by zero, \( x^2 - 4 eq 0 \). This means \( x^2 eq 4 \), or \( x eq \pm 2 \). The domain of the inequality is \( x \in \mathbb{R} \setminus \{-2, 2\} \).
02

Find Critical Points

Solve the equation \( x^2 - 4 = 0 \) to find the critical points which are \( x = \pm 2 \). These points are important because they divide the number line into different intervals.
03

Test Intervals

Test the intervals derived from the critical points \((-\infty, -2)\), \((-2, 2)\), and \((2, \infty)\). Pick a test point from each interval and substitute it into \( \frac{5}{x^2 - 4} \). For example:- For \((-\infty, -2)\), test \(x = -3\) resulting in \( \frac{5}{(-3)^2-4} = \frac{5}{5} = 1 > 0 \).- For \((-2, 2)\), test \(x = 0\) yielding \( \frac{5}{0^2-4} = \frac{5}{-4} < 0 \).- For \((2, \infty)\), test \(x = 3\) resulting in \( \frac{5}{9-4} = \frac{5}{5} = 1 > 0 \).
04

Write the Solution Symbolically

The inequality \( \frac{5}{x^2 - 4} < 0 \) holds in the interval \( (-2, 2) \), as this is where the expression is negative. We exclude \( x = -2 \) and \( x = 2 \) since they are not in the domain.
05

Graphical Interpretation

Plot the function \( y = \frac{5}{x^2 - 4} \). Identify that the graph only dips below the x-axis (where the function is negative) between \( x = -2 \) and \( x = 2 \). The graph approaches vertical asymptotes at \( x = -2 \) and \( x = 2 \).
06

Graphical Solution

Confirm the intervals where the graph of \( \frac{5}{x^2 - 4} \) is below the x-axis (negative). The valid interval for the inequality is \( -2 < x < 2 \), confirming the symbolic solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain Determination
When solving rational inequalities like \( \frac{5}{x^2 - 4} < 0 \), the first step is to determine the domain of the inequality. Rational expressions are undefined when their denominators are zero. So, we need to find where \( x^2 - 4 = 0 \).
Interval Testing
After determining the domain, the next step involves isolating the number line into intervals based on the critical points. The critical points are where \( x^2 - 4 = 0 \), which are \( x = -2 \) and \( x = 2 \).
  • \((-\infty, -2)\)
  • \((-2, 2)\)
  • \((2, \infty)\)
    • For each interval, choose a test point (a random number from the interval) and plug it into \( \frac{5}{x^2 - 4} \) to check the sign of the expression.
  • In \((-\infty, -2)\), for \( x = -3 \), the expression is positive.
  • In \((-2, 2)\), for \( x = 0 \), the expression is negative.
  • In \((2, \infty)\), for \( x = 3 \), the expression is positive.
    • So, the inequality holds only in the interval \((-2, 2)\).
    Graphical Interpretation
    Visualizing the inequality helps understand why the interval \((-2, 2)\) satisfies \( \frac{5}{x^2 - 4} < 0 \). By graphing \( y = \frac{5}{x^2 - 4} \), you can see how the curve behaves at and around the critical points.
    Critical Points
    Critical points for this problem were established by solving \( x^2 - 4 = 0 \), yielding \( x = \pm 2 \). These points are integral in solving the problem because they create boundaries within which the sign of the rational expression can differ.

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