Math Problem Statement

If[x+ [2x ]] <3 where [] denotes the greatest integer function, then

Solution

We are given the inequality x+2x<3x + \left\lfloor 2x \right\rfloor < 3, where y\left\lfloor y \right\rfloor denotes the greatest integer function, meaning the largest integer less than or equal to yy.

Step-by-step solution:

  1. Understanding the greatest integer function:

    The function 2x\left\lfloor 2x \right\rfloor represents the greatest integer less than or equal to 2x2x.

  2. Breaking the problem down:

    We need to find the values of xx that satisfy the inequality: x+2x<3.x + \left\lfloor 2x \right\rfloor < 3.

    To approach this, consider different intervals based on the value of xx, as the floor function 2x\left\lfloor 2x \right\rfloor changes at integer values of 2x2x.

  3. Case analysis:

    Let's look at different intervals for xx:

    • When 0x<0.50 \leq x < 0.5:

      • In this range, 02x<10 \leq 2x < 1, so 2x=0\left\lfloor 2x \right\rfloor = 0.
      • The inequality becomes: x+0<3orx<3.x + 0 < 3 \quad \text{or} \quad x < 3.
      • Since xx is already constrained by 0x<0.50 \leq x < 0.5, the inequality holds for this range.
    • When 0.5x<10.5 \leq x < 1:

      • In this range, 12x<21 \leq 2x < 2, so 2x=1\left\lfloor 2x \right\rfloor = 1.
      • The inequality becomes: x+1<3orx<2.x + 1 < 3 \quad \text{or} \quad x < 2.
      • Since xx is already constrained by 0.5x<10.5 \leq x < 1, the inequality holds for this range.
    • When 1x<1.51 \leq x < 1.5:

      • In this range, 22x<32 \leq 2x < 3, so 2x=2\left\lfloor 2x \right\rfloor = 2.
      • The inequality becomes: x+2<3orx<1.x + 2 < 3 \quad \text{or} \quad x < 1.
      • This condition contradicts 1x<1.51 \leq x < 1.5, so there are no solutions in this range.
    • When 1.5x<21.5 \leq x < 2:

      • In this range, 32x<43 \leq 2x < 4, so 2x=3\left\lfloor 2x \right\rfloor = 3.
      • The inequality becomes: x+3<3orx<0.x + 3 < 3 \quad \text{or} \quad x < 0.
      • This condition contradicts 1.5x<21.5 \leq x < 2, so there are no solutions in this range.
  4. Conclusion:

    The only values of xx that satisfy the inequality are those in the range 0x<10 \leq x < 1.

    Therefore, the solution is: 0x<1.0 \leq x < 1.


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:

  1. How does the greatest integer function impact the solution of quadratic inequalities?
  2. Can the greatest integer function be used in solving rational inequalities?
  3. What are the key properties of the greatest integer function that are useful in algebraic manipulation?
  4. How can we extend this method to inequalities involving higher powers of xx?
  5. 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 2x2x. 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