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 such that the expression
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:
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:
For the denominator:
Step 2: Rewrite the entire expression
Now, the original expression becomes:
Step 3: Set conditions for the expression to be a positive integer
For the fraction to be a positive integer, the denominator must divide the numerator evenly.
Let’s express this as a condition:
For this to be an integer, must be an integer, which means must be a divisor of 56.
Step 4: Find divisors of 56
The divisors of 56 are:
Step 5: Solve for
We now solve , where is a divisor of 56. For each divisor , we solve for :
- If , then , so , which is not an integer.
- If , then , so (integer).
- If , then , so (integer).
We continue testing, but we already see that and are candidates. We now check which one is the least value.
Step 6: Conclusion
The least positive integer value of is .
Would you like me to explain any of the steps in further detail or explore other related questions?
Related questions:
- What happens if you try the divisors of 56 greater than 2?
- What is the general condition for a rational expression to be an integer?
- Can you find a different approach to solve this equation by substituting ?
- How does the expression change if the constant 7/3 were modified?
- 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
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