Problem 57 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The solution is \((-\infty, 0)\).

Step 1: Identify Critical Points

To solve the rational inequality \( \frac{1}{x} < 0 \), first identify the critical points where the expression is undefined or equal to zero. The expression \( \frac{1}{x} \) is undefined at \( x = 0 \). There are no points where it equals zero, as no values of \( x \) make the fraction equal to zero.

Step 2: Determine Intervals

Using the critical point found, \( x = 0 \), divide the number line into intervals. The intervals are \( (-\infty, 0) \) and \( (0, \infty) \).

Step 3: Test Intervals

Choose a test point from each interval and substitute it into the inequality \( \frac{1}{x} < 0 \). For \( x = -1 \) in \( (-\infty, 0) \), \( \frac{1}{-1} = -1 < 0 \), which satisfies the inequality. For \( x = 1 \) in \( (0, \infty) \), \( \frac{1}{1} = 1 > 0 \), which does not satisfy the inequality.

Step 4: Symbolic Solution Conclusion

The solution to the inequality \( \frac{1}{x} < 0 \) is the interval \( (-\infty, 0) \), since that is where the inequality holds true.

Step 5: Graphical Representation

Graph the function \( y = \frac{1}{x} \). Note that the function has a vertical asymptote at \( x=0 \) and is positive for \( x > 0 \) and negative for \( x < 0 \). The inequality \( \frac{1}{x} < 0 \) corresponds to the portion of the graph below the x-axis, which occurs for \( x < 0 \). Thus, graphically, the solution is \( (-\infty, 0) \).

Key Concepts

Critical Points

When tackling rational inequalities, identifying critical points is crucial. For the inequality \( \frac{1}{x} < 0 \), critical points are where the expression becomes undefined or equals zero. Here, the only critical point is at \( x = 0 \) because \( \frac{1}{x} \) becomes undefined at this value as division by zero is not possible. In rational inequalities, critical points signal changes in the inequality's behavior, so noting them helps us to focus on how the inequality behaves across different intervals.

Interval Testing

Interval testing involves dividing the number line into sections based on the critical points. For \( x = 0 \), we split the line into two intervals: \( (-\infty, 0) \) and \( (0, \infty) \). This process helps us analyze which parts of the inequality's expression satisfy the given condition. By selecting test points (like \( x = -1 \) and \( x = 1 \) for their respective intervals), we substitute these into the inequality \( \frac{1}{x} < 0 \).

• In \( (-\infty, 0) \): \( \frac{1}{-1} = -1 < 0 \), confirming the inequality is true.
• In \( (0, \infty) \): \( \frac{1}{1} = 1 > 0 \), showing the inequality is false.

Testing intervals provides a clear path to identifying where the inequality statement holds true across the number line.

Graphical Representation

Graphical representation provides a visual understanding of an inequality. For \( y = \frac{1}{x} \), draw a graph noting that there's a vertical asymptote at \( x = 0 \).This asymptote indicates a division by zero, causing the function to diverge. The graph consists of two separate curves: one in the negative x-plane, decreasing towards the asymptote, and the other in the positive x-plane, increasing towards it.The inequality \( \frac{1}{x} < 0 \) corresponds to the portion of the curve found below the x-axis. This is where the graph shows negative values, precisely on the interval \( (-\infty, 0) \). Graphically checking solutions ensures a deeper understanding by correlating numerical results with visual data.

Symbolic Solution

Solving rational inequalities symbolically involves clear steps to guarantee the solution matches. Here, symbols deliver a concise answer.First, identify the critical points and divide the number line into manageable sections or intervals. As discussed, \( x = 0 \) divides the number line into \( (-\infty, 0) \) and \( (0, \infty) \).When we tested intervals symbolically, only the interval \( (-\infty, 0) \) made \( \frac{1}{x} < 0 \) true. This interval turns out to be the complete solution because in it, \( \frac{1}{x} \) is consistently negative. Symbolically, this solution tells us where the inequality statement stands firm.Using symbols provides clarity and efficiency in expressing solutions derived through understanding and testing the behavior of inequalities across divided intervals.