Solve the Fraction Equation: Find X When -7/(x+4) = x-4

To solve this equation, we follow these steps:

• Step 1: Identify the original equation as $\frac{-7}{x+4} = x-4$.
• Step 2: Eliminate the fraction by multiplying both sides by $x+4$:
  $-7 = (x-4)(x+4)$.
• Step 3: Recognize the right-hand side as a difference of squares:
  $-7 = x^2 - 16$.
• Step 4: Rearrange the equation into a standard quadratic form:
  $x^2 - 16 + 7 = 0$ which simplifies to $x^2 - 9 = 0$.
• Step 5: Factor the quadratic expression:
  $(x-3)(x+3) = 0$.
• Step 6: Use the zero product property to find potential solutions:
  $x - 3 = 0$ which gives $x = 3$, and
  $x + 3 = 0$ which gives $x = -3$.
• Step 7: Validate that neither solution makes the denominator zero, as $x \neq -4$.

Thus, the solutions to the equation $\frac{-7}{x+4} = x-4$ are $\pm3$.