Ask a New Question
Menu

(7)/(x+1) + (3)/(x-1) = (6)/((x+1)(x-1))

First, write the value or values of the variable that make a denominator zero. Then solve the equation.

1 answer

To solve the equation

\[ \frac{7}{x+1} + \frac{3}{x-1} = \frac{6}{(x+1)(x-1)}, \]

we first need to determine the values of \( x \) that make the denominators zero.

Step 1: Find Critical Values

The denominators in the equation are \( x + 1 \) and \( x - 1 \). Setting each denominator equal to zero gives:

  1. \( x + 1 = 0 \)

    • Solving this gives \( x = -1 \).

  2. \( x - 1 = 0 \)

    • Solving this gives \( x = 1 \).

Thus, the values of \( x \) that make a denominator zero are \( x = -1 \) and \( x = 1 \).

Step 2: Solve the Equation

Now, we will solve the equation. To eliminate the fractions, we multiply each term by the common denominator, which is \( (x+1)(x-1) \):

\[ (x+1)(x-1) \left( \frac{7}{x+1} \right) + (x+1)(x-1) \left( \frac{3}{x-1} \right) = (x+1)(x-1) \left( \frac{6}{(x+1)(x-1)} \right). \]

This simplifies to:

\[ 7(x-1) + 3(x+1) = 6. \]

Step 3: Simplify the Equation

Now, let's expand and simplify:

  1. \( 7(x-1) = 7x - 7 \).
  2. \( 3(x+1) = 3x + 3 \).

Combining these, we get:

\[ 7x - 7 + 3x + 3 = 6. \]

Combining like terms:

\[ (7x + 3x) + (-7 + 3) = 6, \] \[ 10x - 4 = 6. \]

Step 4: Isolate \( x \)

Next, we isolate \( x \):

\[ 10x = 6 + 4, \] \[ 10x = 10, \] \[ x = 1. \]

Step 5: Check for Extraneous Solutions

However, \( x = 1 \) is a value that makes the denominator zero in the original equation, which means this solution is not valid.

Conclusion

The valid values of \( x \) that make a denominator zero are \( x = -1 \) and \( x = 1 \), and the equation has no valid solutions as the only solution derived from solving the equation introduces a zero in the denominator.

Thus, there are no solutions to the original equation.