Math Problem Statement
If[x+ [2x ]] <3 where [] denotes the greatest integer function, then
Solution
We are given the inequality , where denotes the greatest integer function, meaning the largest integer less than or equal to .
Step-by-step solution:
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Understanding the greatest integer function:
The function represents the greatest integer less than or equal to .
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Breaking the problem down:
We need to find the values of that satisfy the inequality:
To approach this, consider different intervals based on the value of , as the floor function changes at integer values of .
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Case analysis:
Let's look at different intervals for :
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When :
- In this range, , so .
- The inequality becomes:
- Since is already constrained by , the inequality holds for this range.
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When :
- In this range, , so .
- The inequality becomes:
- Since is already constrained by , the inequality holds for this range.
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When :
- In this range, , so .
- The inequality becomes:
- This condition contradicts , so there are no solutions in this range.
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When :
- In this range, , so .
- The inequality becomes:
- This condition contradicts , so there are no solutions in this range.
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Conclusion:
The only values of that satisfy the inequality are those in the range .
Therefore, the solution is:
Would you like more details on how the greatest integer function affects inequalities, or any other clarification?
Here are some related questions you may find useful:
- How does the greatest integer function impact the solution of quadratic inequalities?
- Can the greatest integer function be used in solving rational inequalities?
- What are the key properties of the greatest integer function that are useful in algebraic manipulation?
- How can we extend this method to inequalities involving higher powers of ?
- What happens if we replace the greatest integer function with a ceiling function?
Tip: To solve inequalities involving the greatest integer function, always break down the problem into cases depending on the integer boundaries for . This allows you to simplify and solve the inequality step by step.
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Math Problem Analysis
Mathematical Concepts
Inequalities
Greatest Integer Function
Floor Function
Formulas
x + ⌊2x⌋ < 3
Theorems
Properties of the greatest integer function
Suitable Grade Level
Grades 9-12