Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Solve each rational inequality by hand. $$\frac{(x+1)^{2}}{x-2} \leq 0$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1] \cup [-1, 2)\).

Step 1: Identify Critical Points

Identify the values of \(x\) that make the expression equal to zero or undefined: \((x+1)^2 = 0\) when \(x = -1\) and \(x - 2 = 0\) when \(x = 2\). Thus, the critical points are \(x = -1\) and \(x = 2\).

Step 2: Determine the Test Intervals

Divide the number line into intervals using the critical points: \((-\infty, -1)\), \([-1, 2)\), and \((2, \infty)\).

Step 3: Test Each Interval

Select test points in each interval and evaluate the inequality \(\frac{(x+1)^2}{x-2} \leq 0\). Use \(x = -2\) in \((-\infty, -1)\), \(x = 0\) in \([-1, 2)\), and \(x = 3\) in \((2, \infty)\).

Step 4: Check Interval \((-\infty, -1)\)

For \(x = -2\), \(\frac{(x+1)^2}{x-2} = \frac{(-2+1)^2}{-2-2} = \frac{1}{-4}<0\). Thus, the inequality holds for \(x \in (-\infty, -1)\).

Step 5: Check Interval \([-1, 2)\)

For \(x = 0\), \(\frac{(x+1)^2}{x-2} = \frac{(0+1)^2}{0-2} = \frac{1}{-2}<0\). The inequality holds for \(x \in [-1, 2)\).

Step 6: Check Interval \((2, \infty)\)

For \(x = 3\), \(\frac{(x+1)^2}{x-2} = \frac{(3+1)^2}{3-2} = \frac{16}{1}>0\). The inequality does not hold for \(x \in (2, \infty)\).

Step 7: Analyze Critical Points

Check the critical points: At \(x = -1\), the expression is defined and equals zero, so it satisfies the inequality. At \(x = 2\), the expression is undefined.

Step 8: Determine Solution Set

The solution is the union of intervals \((-\infty, -1]\) and \([-1, 2)\).

Key Concepts

Critical Points

The concept of critical points is fundamental when solving rational inequalities. Critical points are specific values of the variable, in this case, **x**, where the function either becomes zero or undefined.

For the inequality given, \( \frac{(x+1)^2}{x-2} \leq 0 \), we determine critical points by setting the numerator and denominator equal to zero separately.

**Finding Critical Points:**
\((x+1)^2 = 0\) gives \(x = -1\). Similarly, \((x-2) = 0\) yields \(x = 2\).

These points are crucial because they help us establish divisions on the number line, which we will use to create test intervals. Knowing where the function becomes zero or undefined allows us to analyze and test specific intervals thoroughly.

Test Intervals

Test intervals are sections of the number line determined by critical points. They allow us to evaluate different regions where the inequality might hold.
Test intervals begin at one critical point and extend to the next.

**Creating Test Intervals:**

• The critical points \(x = -1\) and \(x = 2\) divide the number line into intervals: \((-\infty, -1)\), \([-1, 2)\), and \((2, \infty)\).

These intervals help us identify where the inequality is true by testing points within them. It is important to pick test points in each interval that make calculation simple, yet show where the inequality holds.

Without careful selection of test intervals, you may miss parts of the solution set, leading to inaccurate conclusions.

Solution Set Analysis

Solution set analysis involves determining which intervals satisfy the inequality. Once test points are evaluated, we piece together these results to form the solution set.

**Analyzing Intervals:**

• For \((-\infty, -1)\), select \(x = -2\). This gives \(\frac{(x+1)^2}{x-2} = \frac{1}{-4} < 0\), confirming the inequality holds for this interval.
• In \([-1, 2)\), use \(x = 0\). The result \(\frac{1}{-2} < 0\) confirms the inequality holds.
• For \((2, \infty)\), \(x = 3\) yields \(\frac{16}{1} > 0\), which does not satisfy the inequality.

Solution set analysis also checks critical points. For example, \(x = -1\) meets the inequality since it equals zero. However, \(x = 2\) is undefined, meaning it isn't included in the solution set.

These analyses guide in assembling the full solution set accurately.

Inequality Testing

Inequality testing is vital in confirming if certain intervals satisfy the inequality. It involves substituting different test values from each interval into the inequality, following logical evaluations.

**Steps for Testing:**

• Select a representative value from each interval created in the previous critical point assessment.
• Substitute into the inequality \(\frac{(x+1)^2}{x-2} \leq 0\) as done in the step-by-step example.
• Evaluate to determine if the result confirms or disconfirms the inequality.

This testing ensures you include only the intervals that form part of the solution set. It is also crucial in eliminating intervals where the inequality fails to hold.

Inequality testing ensures a thorough and precise approach—vital for solving rational inequalities accurately.