Math Problem Statement

6/(x-1) = x / (x-2)

Solution

To solve the equation 6x1=xx2\frac{6}{x-1} = \frac{x}{x-2}, we can cross-multiply:

6(x2)=x(x1)6(x - 2) = x(x - 1)

Expanding both sides:

6x12=x2x6x - 12 = x^2 - x

Rearranging all terms to one side gives:

x27x+12=0x^2 - 7x + 12 = 0

Next, we can factor this quadratic equation:

(x3)(x4)=0(x - 3)(x - 4) = 0

Thus, the solutions are:

x=3orx=4x = 3 \quad \text{or} \quad x = 4

Make sure to check that these values do not make the denominators zero.

Would you like to explore further details or have any questions? Here are some related questions you might consider:

  1. How can we verify the solutions?
  2. What would happen if we had a different constant on the left side?
  3. Can we solve this using another method, like completing the square?
  4. How do the solutions affect the original equation graphically?
  5. What restrictions exist for xx based on the original denominators?

Tip: Always check your solutions in the original equation to ensure they don’t result in division by zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Equations
Quadratic Equations

Formulas

Cross-multiplication
Factoring quadratic equations

Theorems

Zero Product Property

Suitable Grade Level

Grades 7-9