(4–x):(4+x)=1-5 2/13

To solve the equation \((4 - x):(4 + x) = 1 - 5\frac{2}{13}\), we first convert the mixed number to an improper fraction.

The mixed number \(5\frac{2}{13}\) equals: \[ 5 \times 13 + 2 = 65 + 2 = 67 \] So, \[ 1 - 5\frac{2}{13} = 1 - \frac{67}{13} = \frac{13}{13} - \frac{67}{13} = \frac{13 - 67}{13} = \frac{-54}{13}. \] Now, substituting this back into the equation, we rewrite it as: \[ \frac{4 - x}{4 + x} = \frac{-54}{13}. \]

Next, we cross-multiply to solve for \(x\): \[ (4 - x) \cdot 13 = -54 \cdot (4 + x). \] Expanding both sides gives: \[ 52 - 13x = -216 - 54x. \] Now, let's isolate \(x\) by bringing terms involving \(x\) to one side and constant terms to the other side. Adding \(54x\) and subtracting \(52\) from both sides results in: \[ 54x - 13x = -216 - 52, \] which simplifies to: \[ 41x = -268. \] Now, divide both sides by 41: \[ x = \frac{-268}{41}. \] Calculating this gives: \[ x = -\frac{268}{41} = -6.533... \text{ or approximately } -6.54. \] Thus, the solution is: \[ x \approx -6.54. \]