Math Problem Statement

Solve the rational inequality: (x - 4) / (x + 8) < 0. Express the solution set in interval notation.

Solution

We are asked to solve the rational inequality:

x4x+8<0\frac{x - 4}{x + 8} < 0

Steps to solve:

  1. Find the critical points: These occur where the numerator and denominator are equal to zero.

    • Numerator: x4=0    x=4x - 4 = 0 \implies x = 4
    • Denominator: x+8=0    x=8x + 8 = 0 \implies x = -8
  2. Divide the number line: Use the critical points to divide the number line into intervals:

    • Interval 1: (,8)(-\infty, -8)
    • Interval 2: (8,4)(-8, 4)
    • Interval 3: (4,)(4, \infty)
  3. Test each interval: Pick a test point from each interval and substitute it into the expression x4x+8\frac{x - 4}{x + 8} to determine the sign.

    • For x=9x = -9 (Interval 1: (,8)(-\infty, -8)): 949+8=131=13>0\frac{-9 - 4}{-9 + 8} = \frac{-13}{-1} = 13 > 0 So, the expression is positive in this interval.

    • For x=0x = 0 (Interval 2: (8,4)(-8, 4)): 040+8=48=12<0\frac{0 - 4}{0 + 8} = \frac{-4}{8} = -\frac{1}{2} < 0 So, the expression is negative in this interval.

    • For x=5x = 5 (Interval 3: (4,)(4, \infty)): 545+8=113>0\frac{5 - 4}{5 + 8} = \frac{1}{13} > 0 So, the expression is positive in this interval.

  4. Determine the solution: We need the expression to be less than zero, so we take the interval where it is negative.

    The expression is negative in (8,4)(-8, 4).

  5. Handle the endpoints:

    • x=8x = -8 makes the denominator zero, so it is undefined, and we exclude it.
    • x=4x = 4 makes the numerator zero, but we are looking for strictly less than zero, so exclude it.

Final Solution:

The solution set is:

(8,4)(-8, 4)

Would you like further details on any part of this process?

Here are some related questions:

  1. How do you solve rational inequalities in general?
  2. What happens when the inequality is greater than or equal to 0 instead?
  3. How do you handle more complex rational expressions in inequalities?
  4. Why do we exclude points where the denominator is zero?
  5. Can a rational inequality have more than two critical points?

Tip: Always remember to check for undefined points (where the denominator is zero) in rational inequalities, as they can affect the solution interval.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rational Inequalities
Number Line Analysis
Critical Points

Formulas

Rational Expression: (x - 4) / (x + 8)

Theorems

Sign Analysis of Rational Functions
Critical Point Analysis

Suitable Grade Level

Grades 9-11