To determine which ordered pairs are viable solutions where [tex]\( x \)[/tex] represents the number of days a library book is late and [tex]\( y \)[/tex] represents the total fee charged at a rate of [tex]$0.30 per day, we need to evaluate each pair.
First, let's recall the given fee per day:
\[ \text{Fee per day} = \$[/tex]0.30 \]
We will calculate the total fee [tex]\( y \)[/tex] for each [tex]\( x \)[/tex] by using the equation:
[tex]\[ y = x \times 0.30 \][/tex]
Let's evaluate each ordered pair provided in the question:
1. Ordered pair [tex]\((-3, -0.90)\)[/tex]:
[tex]\[
x = -3, \quad y = -3 \times 0.30 = -0.90
\][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\(-0.90\)[/tex]. While this calculation is correct, note that a viable solution should have a positive [tex]\( x \)[/tex] value since the number of days cannot be negative.
2. Ordered pair [tex]\((-2.5, -0.75)\)[/tex]:
[tex]\[
x = -2.5, \quad y = -2.5 \times 0.30 = -0.75
\][/tex]
Again, the calculated [tex]\( y \)[/tex] value is [tex]\(-0.75\)[/tex]. However, like the previous case, a negative [tex]\( x \)[/tex] value is not viable since it is not possible to have negative days.
3. Ordered pair [tex]\((4.5, 1.35)\)[/tex]:
[tex]\[
x = 4.5, \quad y = 4.5 \times 0.30 = 1.35
\][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\( 1.35 \)[/tex]. This pair is viable since both [tex]\( x \)[/tex] (number of days late) and [tex]\( y \)[/tex] (total fee) are positive and correctly calculated based on the fee per day.
4. Ordered pair [tex]\((8, 2.40)\)[/tex]:
[tex]\[
x = 8, \quad y = 8 \times 0.30 = 2.40
\][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\( 2.40 \)[/tex]. This pair is also viable as both the number of days and the total fee are positive and correctly calculated.
Therefore, the viable solutions are:
[tex]\[
(4.5, 1.35) \quad \text{and} \quad (8, 2.40)
\][/tex]
