To find the equation that models the maximum and minimum amounts of a sports drink advertised as 20 fluid ounces with a variance of 0.45 fluid ounces, we need to understand absolute value inequalities. The absolute value function will help us determine the range of acceptable fluid ounces.
The equation we will form using an absolute value function involves modeling the range of the drink's true volume around a fixed point. Here, the fixed amount is 20 fluid ounces, and it can vary by 0.45 fluid ounces.
The correct equation to represent this scenario is:
[tex]\[ |x - 20| = 0.45 \][/tex]
This equation says that the difference between the actual volume [tex]\( x \)[/tex] and the advertised volume (20 fluid ounces) is 0.45 fluid ounces.
To solve this equation, we consider the definition of absolute value:
[tex]\[ |x - 20| = 0.45 \][/tex]
This splits into two separate equations:
[tex]\[ x - 20 = 0.45 \quad \text{and} \quad x - 20 = -0.45 \][/tex]
Solving each equation for [tex]\( x \)[/tex]:
1. [tex]\( x - 20 = 0.45 \)[/tex]
[tex]\[ x = 20 + 0.45 \][/tex]
[tex]\[ x = 20.45 \][/tex]
2. [tex]\( x - 20 = -0.45 \)[/tex]
[tex]\[ x = 20 - 0.45 \][/tex]
[tex]\[ x = 19.55 \][/tex]
Therefore, the minimum amount for the drink, [tex]\( x \)[/tex], is:
[tex]\[ x = 19.55 \text{ ounces} \][/tex]
Additionally, we can verify this by writing the complete acceptable range of [tex]\( x \)[/tex]:
The equation [tex]\(|x - 20| = 0.45\)[/tex] when solved provides the interval:
[tex]\[ x \in [19.55, 20.45] \][/tex]
Thus, the minimum amount of the sports drink is:
[tex]\[ 19.55 \text{ ounces} \][/tex]
