Problem 60 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The solution to the inequality is \(x \in (-1, 1)\).

Step 1: Identify Critical Points

To solve the inequality \( \frac{x-1}{x+1}<0 \), start by identifying where the expression is equal to zero or undefined. The expression is zero when the numerator is zero, i.e., \( x - 1 = 0 \), giving \( x = 1 \). The expression is undefined when the denominator is zero, i.e., \( x + 1 = 0 \), giving \( x = -1 \). Thus, the critical points are \( x = 1 \) and \( x = -1 \).

Step 2: Test Intervals

Use the critical points to divide the x-axis into intervals: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\). Select a test point from each interval and substitute it into \( \frac{x-1}{x+1} \). For \(x < -1\), choose \(x = -2\) gives a positive result. For \(-1 < x < 1\), choose \(x = 0\) gives a negative result. For \(x > 1\), choose \(x = 2\) gives a positive result.

Step 3: Solution Analysis

The inequality \( \frac{x-1}{x+1} < 0 \) holds true for intervals where the expression is negative. From the testing, the expression is negative in the interval \((-1, 1)\). Thus, the solution to the inequality is \( x \in (-1, 1) \).

Step 4: Graphical Solution

To solve graphically, plot \( y = \frac{x-1}{x+1} \). Observe where the graph is below the x-axis (\(y < 0\)). The graph dips below the x-axis between \(x = -1\) and \(x = 1\), verifying that \(x \in (-1, 1)\) is the solution.

Key Concepts

Critical Points

When solving rational inequalities, identifying critical points is key. Critical points help us understand where an expression changes its behavior, such as becoming zero or undefined. For the inequality \( \frac{x-1}{x+1}<0 \), we first find where the numerator and denominator are zero. The numerator, \(x-1\), equals zero at \(x = 1\). The denominator, \(x+1\), equals zero at \(x = -1\). So, the critical points are at \(x = 1\) and \(x = -1\). These points are important because they help us determine the boundaries of our intervals, and they're not part of the solution because they cause the expression to be zero or undefined.

Test Intervals

After identifying critical points, the next step is to divide the number line into intervals. These are called test intervals. You use critical points to create test intervals: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\). Each interval is then tested with a sample point. For example:

• In \((-\infty, -1)\), try \(x = -2\)
• In \((-1, 1)\), try \(x = 0\)
• In \((1, \infty)\), try \(x = 2\)

By substituting these points into the expression \( \frac{x-1}{x+1} \), you can determine the sign (positive or negative) of the expression within each interval. This tells us where the inequality holds true.

Graphical Solution

Graphically solving a rational inequality involves plotting the function and observing where it meets certain criteria. For \( \frac{x-1}{x+1}<0 \), you would plot the function \( y = \frac{x-1}{x+1} \) on a graph. The graph reveals where the function is negative, i.e., below the x-axis. In this scenario, the graph dips below the x-axis between \( x = -1 \) and \( x = 1 \). Here, the function value is negative, confirming the solution \( x \in (-1, 1) \). Graphical solutions provide a visual verification, helping you to easily see where inequalities are satisfied.

Symbolic Solution

The symbolic solution is the algebraic approach to solving rational inequalities. For \( \frac{x-1}{x+1}<0 \), you inspect each interval created by the critical points to see if they satisfy the inequality. Use test points within these intervals:

• When \(x = -2\), \( \frac{-3}{-1} = 3\) – positive
• When \(x = 0\), \( \frac{-1}{1} = -1\) – negative
• When \(x = 2\), \( \frac{1}{3} \) – positive

The original inequality \( \frac{x-1}{x+1}<0 \) is true where the result is negative, which occurs in the interval \((-1, 1)\). This method confirms the solution by checking each mathematical region for validity, ensuring that each step is logically sound and mathematically correct.