Problem 39 - Precalculus : Building Concepts and Connections

Question

Solve the rational inequality. $$\frac{x-1}{x+2} \geq 0$$

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{x-1}{x+2} \geq 0\) is \((-∞,-2] \cup [1,∞)\)

Step 1: Identify and Isolate the Critical Points

Find the critical points by setting the numerator and denominator equal to zero separately. \nFor the numerator x-1 = 0, we get x=1. \nFor the denominator x+2 = 0, we get x=-2. These values are the x-values where the function changes its sign or is undefined.

Step 2: Construct the Sign Chart

A sign chart is used to determine where a function is positive or negative. To create the sign chart, list the critical points in ascending order on a number line. Divide the number line into intervals using the critical points. These intervals are (-∞,-2], [-2, 1], and [1,∞).

Step 3: Test each Interval

Select a test point from each interval and substitute these points into the inequality to determine whether the function is positive or negative. \n\nFor the interval (-∞,-2], choose x=-3: \[ \frac{(-3)-1}{(-3)+2}=\frac{-4}{-1}=4>0\], hence this interval is satisfied.\n\nFor the interval [-2, 1], choose x=0: \[ \frac{0-1}{0+2}=-\frac{1}{2}<0\], hence this interval is not satisfied. \n\nFor the interval [1,∞), choose x=2: \[ \frac{2-1}{2+2}=\frac{1}{4}>0], hence this interval is satisfied.

Step 4: Write the Solution in Interval Notation

The function is greater than or equal to zero when x is either less than or equal to -2, or greater than or equal to 1. Therefore, the solution to this inequality in interval notation should be: \((-∞,-2] \cup [1,∞)\)

Key Concepts

Critical Points

When solving rational inequalities like \( \frac{x-1}{x+2} \geq 0 \), one important concept is identifying the critical points. These are x-values where the function equals zero or becomes undefined. To find them, set both the numerator and denominator equal to zero separately.

• Set the numerator \( x - 1 = 0 \) to find \( x = 1 \).
• Set the denominator \( x + 2 = 0 \) to find \( x = -2 \).

These values are vital because they pinpoint where the sign of the function might change. Critical points like \( x = 1 \) result from zeroing the numerator, suggesting a potential sign change when crossing this point. In contrast, \( x = -2 \) arises when the fraction becomes undefined, indicating a vertical asymptote or boundary. Using critical points effectively allows us to analyze the graphical behavior of the inequality around these key values.

Sign Chart

A sign chart is a useful tool to visually map the positivity and negativity of a function over different intervals segregated by our critical points. For the inequality \( \frac{x-1}{x+2} \geq 0 \), we use the critical points to create distinct regions on a number line.

• List the critical points in ascending order: \(-2\) and \(1\).
• Divide the number line into intervals: \((-\infty, -2] \), \([-2, 1] \), and \([1, \infty)\).

Next, we pick a test point within each interval to determine the sign of the function in that interval.* For \((-\infty, -2]\), choosing \( x = -3 \), the function is positive.* For \([-2, 1]\), choosing \( x = 0 \), the function is negative.* For \([1, \infty)\), choosing \( x = 2 \), the function is positive.
By constructing and analyzing a sign chart, we gain insights into where the function satisfies the inequality, making it an indispensable tool in solving rational inequalities.

Interval Notation

Interval notation is a concise way to express the solution to an inequality. For the problem \( \frac{x-1}{x+2} \geq 0 \), we determined through the sign chart that the solution is valid in certain intervals. In interval notation, we include only the intervals where the inequality holds.

• The function is non-negative (satisfies \( \geq 0 \)) in the intervals \((-\infty, -2] \) and \([1, \infty)\).
• Use a union symbol \(\cup\) to join these intervals: \((-\infty, -2] \cup [1, \infty)\).

This format is efficient and clear. It indicates which values of \( x \) satisfy the inequality without unnecessary detail. Interval notation captures both endpoints where inequality holds inclusive if the inequality is non-strict, indicated by square brackets. An important takeaway is that this notation is useful for presenting solutions succinctly and accurately in mathematical contexts.