To solve the equation
\[ 5 - \sqrt{3 - x} = -2, \]
let's isolate the square root term:
- Subtract 5 from both sides:
\[ -\sqrt{3 - x} = -2 - 5, \] \[ -\sqrt{3 - x} = -7. \]
- Multiply both sides by -1:
\[ \sqrt{3 - x} = 7. \]
- Now, square both sides to eliminate the square root:
\[ 3 - x = 49. \]
- Rearranging gives:
\[ -x = 49 - 3, \] \[ -x = 46. \]
- Multiply both sides by -1 to solve for \( x \):
\[ x = -46. \]
Now, let’s check if this value satisfies the original equation:
Substituting \( x = -46 \) back into the original equation:
\[ 5 - \sqrt{3 - (-46)} = 5 - \sqrt{3 + 46} = 5 - \sqrt{49} = 5 - 7 = -2. \]
This is true!
Thus, the solution to the equation is
\[ \boxed{-46}. \]