Math Problem Statement
Solution
We are asked to solve the rational inequality:
Steps to solve:
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Find the critical points: These occur where the numerator and denominator are equal to zero.
- Numerator:
- Denominator:
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Divide the number line: Use the critical points to divide the number line into intervals:
- Interval 1:
- Interval 2:
- Interval 3:
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Test each interval: Pick a test point from each interval and substitute it into the expression to determine the sign.
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For (Interval 1: ): So, the expression is positive in this interval.
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For (Interval 2: ): So, the expression is negative in this interval.
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For (Interval 3: ): So, the expression is positive in this interval.
-
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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 .
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Handle the endpoints:
- makes the denominator zero, so it is undefined, and we exclude it.
- makes the numerator zero, but we are looking for strictly less than zero, so exclude it.
Final Solution:
The solution set is:
Would you like further details on any part of this process?
Here are some related questions:
- How do you solve rational inequalities in general?
- What happens when the inequality is greater than or equal to 0 instead?
- How do you handle more complex rational expressions in inequalities?
- Why do we exclude points where the denominator is zero?
- 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.
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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