Certainly! Let's work through the problem by analyzing the given equation and determining what [tex]\( x \)[/tex] represents.
The given equation is:
[tex]\[ 18 + 3x = 24 + 2x \][/tex]
Here:
- [tex]\( 18 \)[/tex] represents the initial cost of the rectangular bird feeder.
- [tex]\( 3x \)[/tex] represents the ongoing cost to maintain the rectangular bird feeder over [tex]\( x \)[/tex] weeks, where [tex]\( x \)[/tex] is the number of weeks.
- [tex]\( 24 \)[/tex] represents the initial cost of the cylindrical bird feeder.
- [tex]\( 2x \)[/tex] represents the ongoing cost to maintain the cylindrical bird feeder over [tex]\( x \)[/tex] weeks, where [tex]\( x \)[/tex] is the number of weeks.
We are asked to determine the value of [tex]\( x \)[/tex] to find the break-even point between the two bird feeders' total costs over time.
Let's solve the equation step by step:
1. Start with the equation:
[tex]\[ 18 + 3x = 24 + 2x \][/tex]
2. Subtract [tex]\( 2x \)[/tex] from both sides to get all [tex]\( x \)[/tex]-terms on one side:
[tex]\[ 18 + 3x - 2x = 24 + 2x - 2x \][/tex]
[tex]\[ 18 + x = 24 \][/tex]
3. Subtract 18 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ 18 + x - 18 = 24 - 18 \][/tex]
[tex]\[ x = 6 \][/tex]
The solution [tex]\( x = 6 \)[/tex] represents the number of weeks at which both bird feeders will have the same total cost. Therefore, [tex]\( x \)[/tex] is:
[tex]\[ \text{The number of weeks the bird feeders are filled.} \][/tex]
Hence, the correct interpretation of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{\text{the number of weeks the bird feeders are filled}} \][/tex]
