Math Problem Statement

What is the least positive integer value of x such that (x/4+7/3)/(x/4−7/3) is a positive integer?

Solution

We are tasked with finding the least positive integer value of xx such that the expression

x4+73x473\frac{\frac{x}{4} + \frac{7}{3}}{\frac{x}{4} - \frac{7}{3}}

is a positive integer.

Step 1: Simplify the expression

Let’s rewrite the given expression in a simpler form. First, combine the terms in both the numerator and denominator:

x4+73x473.\frac{\frac{x}{4} + \frac{7}{3}}{\frac{x}{4} - \frac{7}{3}}.

To simplify this, we’ll first get a common denominator for the terms inside both the numerator and denominator. In both cases, the common denominator is 12.

For the numerator: x4+73=3x12+2812=3x+2812.\frac{x}{4} + \frac{7}{3} = \frac{3x}{12} + \frac{28}{12} = \frac{3x + 28}{12}.

For the denominator: x473=3x122812=3x2812.\frac{x}{4} - \frac{7}{3} = \frac{3x}{12} - \frac{28}{12} = \frac{3x - 28}{12}.

Step 2: Rewrite the entire expression

Now, the original expression becomes:

3x+28123x2812=3x+283x28.\frac{\frac{3x + 28}{12}}{\frac{3x - 28}{12}} = \frac{3x + 28}{3x - 28}.

Step 3: Set conditions for the expression to be a positive integer

For the fraction 3x+283x28\frac{3x + 28}{3x - 28} to be a positive integer, the denominator 3x283x - 28 must divide the numerator 3x+283x + 28 evenly.

Let’s express this as a condition:

3x+283x28=1+563x28.\frac{3x + 28}{3x - 28} = 1 + \frac{56}{3x - 28}.

For this to be an integer, 563x28\frac{56}{3x - 28} must be an integer, which means 3x283x - 28 must be a divisor of 56.

Step 4: Find divisors of 56

The divisors of 56 are:

±1,±2,±4,±7,±8,±14,±28,±56.\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56.

Step 5: Solve for xx

We now solve 3x28=d3x - 28 = d, where dd is a divisor of 56. For each divisor dd, we solve for xx:

  • If 3x28=13x - 28 = 1, then 3x=293x = 29, so x=293x = \frac{29}{3}, which is not an integer.
  • If 3x28=23x - 28 = 2, then 3x=303x = 30, so x=10x = 10 (integer).
  • If 3x28=13x - 28 = -1, then 3x=273x = 27, so x=9x = 9 (integer).

We continue testing, but we already see that x=9x = 9 and x=10x = 10 are candidates. We now check which one is the least value.

Step 6: Conclusion

The least positive integer value of xx is 9\boxed{9}.

Would you like me to explain any of the steps in further detail or explore other related questions?

Related questions:

  1. What happens if you try the divisors of 56 greater than 2?
  2. What is the general condition for a rational expression to be an integer?
  3. Can you find a different approach to solve this equation by substituting y=x4y = \frac{x}{4}?
  4. How does the expression change if the constant 7/3 were modified?
  5. How can you solve a similar problem if you change the denominator?

Tip:

Always factor the numerator and denominator completely to simplify expressions when possible.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Divisibility

Formulas

\frac{\frac{x}{4} + \frac{7}{3}}{\frac{x}{4} - \frac{7}{3}}
Divisibility Rule: numerator and denominator must divide evenly for the result to be an integer.

Theorems

Divisibility Theorem
Simplification of Rational Expressions

Suitable Grade Level

Grades 9-12