Solve and Apply: Finding the Field of Application for (x+y:3)/(2x+6)=4

To solve this problem, we'll follow these steps to find the domain:

• Step 1: Recognize that the expression $\frac{x+y:3}{2x+6}=4$ involves a fraction. The denominator $2x + 6$ must not be zero, as division by zero is undefined.
• Step 2: Set the denominator equal to zero and solve for $x$ to find the values that must be excluded: $2x + 6 = 0$.
• Step 3: Solve $2x + 6 = 0$:
• $2x + 6 = 0$
• $2x = -6$
• $x = -3$
• Step 4: Conclude that the domain of the function excludes $x = -3$, meaning $x \neq -3$.

Thus, the domain of the given expression is all real numbers except $x = -3$. This translates to:

$x\operatorname{\ne}-3$