Problem 61

Question

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

Step-by-Step Solution

Verified

Answer

The solution is $(-2, 2)$.

1Step 1: 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\} $.

2Step 2: 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.

3Step 3: 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 $.

4Step 4: 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.

5Step 5: 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 $.

6Step 6: 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.

Key Concepts

Domain DeterminationInterval TestingGraphical InterpretationCritical Points

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.

Problem 61

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