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(1+1/x+1)(1-1/x-1)=7/8

2 min read Jun 16, 2024
(1+1/x+1)(1-1/x-1)=7/8

Solving the Equation: (1 + 1/(x+1))(1 - 1/(x-1)) = 7/8

This problem presents us with an equation involving fractions and variables. Let's break down the steps to solve for the value of 'x'.

1. Simplify the equation:

  • Combine terms within the parentheses:

    (1 + 1/(x+1)) can be rewritten as (x+1+1)/(x+1) = (x+2)/(x+1)

    (1 - 1/(x-1)) can be rewritten as (x-1-1)/(x-1) = (x-2)/(x-1)

  • Substitute the simplified terms back into the equation:

    [(x+2)/(x+1)] * [(x-2)/(x-1)] = 7/8

2. Multiply both sides of the equation:

  • Multiply the numerators and denominators on the left side:

    (x+2)(x-2) / (x+1)(x-1) = 7/8

  • Cross-multiply to eliminate the fractions:

    8(x+2)(x-2) = 7(x+1)(x-1)

3. Expand and simplify the equation:

  • Expand the products on both sides:

    8(x² - 4) = 7(x² - 1)

    8x² - 32 = 7x² - 7

  • Move all terms to one side:

    8x² - 7x² - 32 + 7 = 0

    x² - 25 = 0

4. Solve for 'x':

  • Factor the quadratic equation:

    (x + 5)(x - 5) = 0

  • Set each factor equal to zero and solve:

    x + 5 = 0 => x = -5 x - 5 = 0 => x = 5

Conclusion:

Therefore, the solutions to the equation (1 + 1/(x+1))(1 - 1/(x-1)) = 7/8 are x = -5 and x = 5.

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