Sure, let's consider the given problem step by step.
### Part A: Writing the absolute value equation
The car travels an average of 28 miles per gallon, but this value can vary by 8 miles per gallon. To represent this situation using an absolute value equation, let's denote [tex]\( x \)[/tex] as the actual miles per gallon the car can achieve.
The absolute value equation to represent the range of miles the car could travel per gallon of gas is given by:
[tex]\[ |x - 28| \leq 8 \][/tex]
This equation states that the deviation of [tex]\( x \)[/tex] from the average value of 28 miles per gallon does not exceed 8 miles per gallon.
### Part B: Solving the equation and interpreting the result
To solve the absolute value equation [tex]\( |x - 28| \leq 8 \)[/tex], we need to break it down into two inequalities:
1. [tex]\( x - 28 \leq 8 \)[/tex]
2. [tex]\( x - 28 \geq -8 \)[/tex]
Let's solve these inequalities separately.
1. Solving the first inequality:
[tex]\[ x - 28 \leq 8 \][/tex]
Adding 28 to both sides:
[tex]\[ x \leq 36 \][/tex]
2. Solving the second inequality:
[tex]\[ x - 28 \geq -8 \][/tex]
Adding 28 to both sides:
[tex]\[ x \geq 20 \][/tex]
Combining these results, we get the range of miles per gallon that the car can achieve:
[tex]\[ 20 \leq x \leq 36 \][/tex]
### Interpretation in terms of the real-world scenario
The range [tex]\( 20 \leq x \leq 36 \)[/tex] implies that, depending on driving conditions and habits, the car can travel between 20 and 36 miles per gallon of gas. This means the car's fuel efficiency won't go below 20 miles per gallon and won't exceed 36 miles per gallon under varying driving circumstances.
