Math Problem Statement
solve symbolically
Solution
To solve the inequality
follow these steps:
Step 1: Understand the inequality
The inequality will hold when the numerator is non-negative, and the denominator does not cause division by zero.
- The numerator is always positive.
- The denominator will determine the sign of the fraction.
Step 2: Analyze the denominator
The denominator, , affects the sign of the fraction:
- When , the fraction is positive.
- When , the fraction is undefined.
- When , the fraction is negative.
Step 3: Solve
Step 4: Write the solution
- For , the fraction is positive, so is true.
- For , the fraction is undefined.
- For , the fraction is negative, so the inequality is false.
Final Solution:
This is the solution in interval notation.
Would you like a detailed explanation of any step?
Here are 5 follow-up questions:
- How does the inequality change if ?
- Can we include in the solution?
- What happens if the numerator also depends on ?
- How would this inequality change if the fraction were instead?
- What are some applications of such inequalities in real-life scenarios?
Tip: Always check for division by zero when solving rational inequalities.
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Math Problem Analysis
Mathematical Concepts
Inequalities
Rational Expressions
Formulas
Sign analysis of rational expressions
Interval notation
Theorems
Properties of rational functions
Sign chart method for inequalities
Suitable Grade Level
Grades 10-12
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