Math Problem Statement
solve the inequality 0 < (2x-1)/(x+1) < 2.
Solution
To solve the inequality , we can break it into two inequalities:
Let's solve these step by step.
Step 1: Solve
For this inequality, we need , which implies that the numerator and the denominator must have the same sign (either both positive or both negative).
Case 1: Both numerator and denominator are positive
So, for both to be positive, we need .
Case 2: Both numerator and denominator are negative
Thus, for both to be negative, we need .
So, the solution to the first inequality is:
Step 2: Solve
To solve , we start by isolating on one side of the inequality: Simplify by writing the left side with a common denominator: Expanding and combining like terms in the numerator:
For , the inequality holds when , meaning .
Step 3: Combine Results
Now we combine the results from both inequalities:
- From , we found or .
- From , we found .
The solution is the intersection of these two results, which is:
Final Answer
The solution to the inequality is:
Would you like more details on any part of this solution?
Here are some related questions that may help expand your understanding:
- How do we solve inequalities involving rational expressions?
- What are the conditions for a rational expression to be positive or negative?
- How do we combine solutions from multiple inequalities?
- Can we solve similar inequalities with quadratic expressions?
- What happens if we change the inequality bounds (for example, )?
Tip: Always check the domain of the rational expression first, as some values may be undefined due to division by zero.
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Math Problem Analysis
Mathematical Concepts
Inequalities
Rational Expressions
Algebra
Formulas
Rational inequality: a/b < c/d
Theorems
Solving rational inequalities
Sign analysis of rational expressions
Suitable Grade Level
Grades 9-11