Problem 46 - Intermediate Algebra

Question

Solve each inequality. $$ \frac{3-x}{x+4} \leq 0 $$

Step-by-Step Solution

Answer

The solution is \( (-\infty, -4) \cup [3, \infty) \).

Step 1: Identify the critical points

The inequality \( \frac{3-x}{x+4} \leq 0 \) can change signs at points where the numerator or the denominator is equal to zero. Set both numerator and denominator to zero separately: 1. Numerator: \( 3-x = 0 \) gives \( x = 3 \).2. Denominator: \( x+4 = 0 \) gives \( x = -4 \).These are the critical points \( x = 3 \) and \( x = -4 \). Divide the real line into intervals with these points.

Step 2: Determine sign of expression on intervals

Evaluate the sign of the expression \( \frac{3-x}{x+4} \) over the intervals created by the critical points: 1. Interval \( (-\infty, -4) \): Choose \( x = -5 \), then \( \frac{3-(-5)}{-5+4} = \frac{8}{-1} = -8 \), which is negative.2. Interval \( (-4, 3) \): Choose \( x = 0 \), then \( \frac{3-0}{0+4} = \frac{3}{4} \), which is positive.3. Interval \( (3, \infty) \): Choose \( x = 4 \), then \( \frac{3-4}{4+4} = \frac{-1}{8} \), which is negative.

Step 3: Analyze inequality sign

Since we want \( \frac{3-x}{x+4} \leq 0 \), select intervals where the expression is negative or zero:- The expression is negative in the intervals \((-\infty, -4)\) and \((3, \infty)\).- The expression equals zero at \( x = 3 \). Thus, the solution includes these intervals and the point \( x = 3 \).

Step 4: Consider open and closed intervals

The inequality \( \frac{3-x}{x+4} \leq 0 \) becomes zero at \( x = 3 \), which will be part of the solution set. However, \( x = -4 \) makes the denominator zero and thus should not be included. Therefore, the solution in interval notation is \( (-\infty, -4) \cup [3, \infty) \).

Key Concepts

Critical Points

Critical points play a vital role when solving algebraic inequalities, especially those involving fractions. For the given inequality \( \frac{3-x}{x+4} \leq 0 \), the process begins by identifying points where the expression can potentially change its sign. These are typically where the numerator or denominator becomes zero.

• Set the numerator, \( 3-x = 0 \), which results in \( x = 3 \).
• Set the denominator, \( x+4 = 0 \), which results in \( x = -4 \).

These calculations reveal the critical points at \( x = 3 \) and \( x = -4 \). It's important to remember that while both points are critical for sign analysis, domains of rational expressions exclude values that make the denominator zero. These critical points help divide the real number line into distinct intervals that can be analyzed separately.

Interval Notation

Interval notation is a standardized way of representing ranges of values on the number line. In this exercise, once critical points have been identified, interval notation is used to express where the inequality \( \frac{3-x}{x+4} \leq 0 \) holds true. Understanding interval notation is crucial for conveying valid solution sets succinctly.

Here, the critical points \( x = 3 \) and \( x = -4 \) divide the number line into three intervals:

• \((-\infty, -4)\)
• \((-4, 3)\)
• \((3, \infty)\)

When writing these intervals, it is important to use round brackets \(()\) for numbers that are not included in the solution (like where the denominator equals zero), and square brackets \([]\) for numbers that are included (solutions to the inequality). For this inequality, the solution in interval notation is expressed as \((-\infty, -4) \cup [3, \infty)\).

Sign Analysis

Sign analysis helps determine where an inequality holds or changes its behavior on the number line. For the inequality \( \frac{3-x}{x+4} \leq 0 \), sign analysis involves testing the expression in the intervals formed by the critical points:

• For \((-\infty, -4)\), substitute a test point like \( x = -5 \): The result is negative, indicating this interval satisfies the inequality.
• For \((-4, 3)\), substitute \( x = 0 \): The result is positive, so it does not satisfy the inequality.
• For \((3, \infty)\), substitute \( x = 4 \): The result is negative, so this interval also satisfies the inequality.

Sign analysis confirms which parts of the number line meet the specified inequality condition. It’s crucial in deciding whether to include or exclude the boundary points. In our case, \(x = 3\) makes the expression zero, which fits \(\leq 0\), while \(x = -4\) must be excluded due to the division by zero risk, refining the solution to \((-\infty, -4) \cup [3, \infty)\). This process solidifies understanding of how solutions comply with inequality conditions.