(x+3)/(x-3)+(x+2)/(x-2)=2

3 min read Jun 16, 2024

Solving the Equation: (x+3)/(x-3) + (x+2)/(x-2) = 2

This equation involves rational expressions and requires careful manipulation to solve for the unknown variable 'x'. Here's a step-by-step solution:

1. Find a Common Denominator

The first step is to find a common denominator for the two fractions on the left side of the equation. The least common denominator is (x-3)(x-2). We can rewrite each fraction with this denominator:

(x+3)/(x-3) * (x-2)/(x-2) = (x+3)(x-2) / (x-3)(x-2)
(x+2)/(x-2) * (x-3)/(x-3) = (x+2)(x-3) / (x-3)(x-2)

2. Combine the Fractions

Now that both fractions have the same denominator, we can combine them:

(x+3)(x-2) + (x+2)(x-3) / (x-3)(x-2) = 2

3. Simplify the Numerator

Expand the numerator by multiplying the binomials:

(x² + x - 6) + (x² - x - 6) / (x-3)(x-2) = 2

Combine like terms:

(2x² - 12) / (x-3)(x-2) = 2

4. Eliminate the Denominator

Multiply both sides of the equation by the denominator to eliminate it:

2x² - 12 = 2(x-3)(x-2)

5. Expand and Solve the Equation

Expand the right side of the equation and simplify:

2x² - 12 = 2(x² - 5x + 6) 2x² - 12 = 2x² - 10x + 12

Subtract 2x² from both sides and add 10x to both sides:

10x = 24

Divide both sides by 10:

x = 2.4

6. Checking for Extraneous Solutions

It's important to check the solution by plugging it back into the original equation. We must ensure that the solution doesn't make any of the denominators zero. In this case, substituting x = 2.4 does not make any denominators zero, so it is a valid solution.

Therefore, the solution to the equation (x+3)/(x-3) + (x+2)/(x-2) = 2 is x = 2.4.