To find the value of \( x \), we'll start by using the formula for the perimeter of a rectangle, which is given by:
\[ P = 2(l + w) \]
where \( l \) is the length and \( w \) is the width. In this case, we have:
- Length \( l = 2x + 12 \) ft
- Width \( w = 5x + 22 \) ft
According to the problem, the perimeter \( P \) is 110 ft. Therefore, we can set up the equation:
\[ 2((2x + 12) + (5x + 22)) = 110 \]
Let's simplify the equation step by step:
- Combine the expressions inside the parentheses:
\[ 2(2x + 12 + 5x + 22) = 110 \] \[ 2(7x + 34) = 110 \]
- Divide both sides by 2:
\[ 7x + 34 = 55 \]
- Subtract 34 from both sides:
\[ 7x = 55 - 34 \] \[ 7x = 21 \]
- Now, divide by 7:
\[ x = \frac{21}{7} = 3 \]
Thus, the value of \( x \) is:
\[ \boxed{3} \text{ ft} \]