Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Solve each rational inequality by hand. $$\frac{2 x-5}{x^{2}-1} \geq 0$$

Step-by-Step Solution

Answer

The solution is \( x \in \left[ \frac{5}{2}, \infty \right) \).

Step 1: Find Critical Points

The inequality is \( \frac{2x-5}{x^2-1} \geq 0 \). Begin by identifying critical points where the fraction is undefined or equals zero. These points occur where the numerator is zero and the denominator is zero. Set the numerator equal to zero: \( 2x - 5 = 0 \) which gives \( x = \frac{5}{2} \). Set the denominator equal to zero: \( x^2 - 1 = 0 \), which factors to \( (x - 1)(x + 1) = 0 \). This gives \( x = 1 \) and \( x = -1 \). So the critical points are \( x = -1, 1, \) and \( \frac{5}{2} \).

Step 2: Test Intervals

Divide the number line into intervals based on the critical points: \((-\infty, -1)\), \((-1, 1)\), \((1, \frac{5}{2})\), and \((\frac{5}{2}, \infty)\). Choose test points from each interval to determine the sign of the expression.

Step 3: Evaluate Sign of Expression in Each Interval

For each interval, plug a test point into the expression \( \frac{2x-5}{x^2-1} \):- Test point \( x = -2 \) in \((-\infty, -1)\) gives a negative value.- Test point \( x = 0 \) in \((-1, 1)\) gives a negative value.- Test point \( x = 2 \) in \((1, \frac{5}{2})\) gives a positive value.- Test point \( x = 3 \) in \((\frac{5}{2}, \infty)\) gives a positive value.

Step 4: Consider Critical Points

Evaluate the expression at the critical points:- At \( x = -1 \) and \( x = 1 \), the expression is undefined because the denominator is zero.- At \( x = \frac{5}{2} \), the expression is zero as the numerator is zero.

Step 5: Determine Solution Set

Include intervals where the expression is positive or zero, complying with the \( \geq 0 \) part of the inequality. Thus, the solution set is \( x \in \left[ \frac{5}{2}, \infty \right) \).

Key Concepts

Critical Points in Rational Inequalities

In solving rational inequalities, the first step is identifying critical points. Critical points are values of the variable that make the rational expression either zero or undefined. To find these, focus on both the numerator and the denominator of the rational expression.
For the equation \( \frac{2x-5}{x^2-1} \geq 0 \), set the numerator \(2x-5\) equal to zero to find where the entire expression equals zero. This gives the solution \(x = \frac{5}{2}\).
Next, find where the denominator \(x^2-1\) equals zero, as these are points where the expression is undefined. When you solve \((x - 1)(x + 1) = 0\), you find the critical points \(x = 1\) and \(x = -1\).

• Critical points where the expression equals zero help identify boundary values for the inequality.
• Points where the expression is undefined identify divisions in the solution set but cannot be part of it.

These critical points help in segmenting the number line into different intervals for testing what happens between these key values.

Solving Rational Inequalities

After identifying critical points, the next step is to determine where the expression satisfies the inequality. In this case, you need to find when \( \frac{2x-5}{x^2-1} \) is greater than or equal to zero.
This involves evaluating the expression within the intervals formed by the critical points. By testing values within each interval, you can determine the sign of the expression.

• If the expression is positive within an interval, it satisfies the condition of being \( \geq 0 \).
• If negative, it does not meet the inequality requirements.

This approach allows identification of valid solution intervals for the inequality.

Using Test Intervals

Test intervals are small sections of the number line between the critical points. These intervals help check the sign of a rational expression throughout the range of possible solutions without calculating every possible value.
Given the critical points \(-1, 1,\) and \(\frac{5}{2}\), the number line is split into intervals: \((-\infty, -1)\), \((-1, 1)\), \((1, \frac{5}{2})\), and \((\frac{5}{2}, \infty)\).
For each interval, you choose a test point, which is an arbitrary number within that section, and plug it into the original inequality \( \frac{2x-5}{x^2-1} \) to check whether it is positive or negative.

• A negative result means the interval does not satisfy the inequality \( \geq 0 \).
• A positive result does satisfy it.

This method ensures you accurately determine the intervals that compose the solution set without uncertainty.