To solve the problem and determine what [tex]\( x \)[/tex] represents in the given equation [tex]\( 18 + 3x = 24 + 2x \)[/tex], we'll follow these steps:
1. Understand the Equation:
- [tex]\( 18 + 3x \)[/tex]: The total cost of the rectangular bird feeder, which is the initial cost of [tex]$\$[/tex]18.00[tex]$ plus \(\$[/tex]3.00\) per week.
- [tex]\( 24 + 2x \)[/tex]: The total cost of the cylindrical bird feeder, which is the initial cost of [tex]$\$[/tex]24.00[tex]$ plus \(\$[/tex]2.00\) per week.
2. Set the Totals Equal:
- The equation [tex]\( 18 + 3x = 24 + 2x \)[/tex] represents the point where the total costs of both bird feeders are equal.
3. Isolate [tex]\( x \)[/tex] to Find the Break-Even Point:
- Start by subtracting [tex]\( 2x \)[/tex] from both sides of the equation to simplify:
[tex]\[
18 + 3x - 2x = 24 + 2x - 2x
\][/tex]
which simplifies to:
[tex]\[
18 + x = 24
\][/tex]
- Next, subtract 18 from both sides to further isolate [tex]\( x \)[/tex]:
[tex]\[
18 + x - 18 = 24 - 18
\][/tex]
which simplifies to:
[tex]\[
x = 6
\][/tex]
4. Interpret [tex]\( x \)[/tex]:
- The solution [tex]\( x = 6 \)[/tex] means that it will take 6 weeks for the total costs of filling both bird feeders to be equal.
Therefore, [tex]\( x \)[/tex] represents the number of weeks the bird feeders are filled before their total costs equalize. Thus, the correct interpretation is:
- The number of weeks the bird feeders are filled.
