Alright, let's analyze the problem and solve it step by step.
The athlete can lift an average of 325 pounds, but this weight can vary by 20 pounds. This means that the weight [tex]\( w \)[/tex] the athlete can lift can be 20 pounds more or 20 pounds less than 325.
To find the equation that represents this scenario, we need to express the variation in terms of absolute value. Absolute value measures the distance from a fixed point, in this case, 325 pounds.
### Step-by-Step Solution:
1. Identify the Fixed Point and the Variation:
- The fixed point is the average weight, which is 325 pounds.
- The variation is ±20 pounds.
2. Formulate the Absolute Value Equation:
- The weight [tex]\( w \)[/tex] can be either 20 pounds more or 20 pounds less than 325 pounds.
- This can be expressed as the absolute difference between [tex]\( w \)[/tex] and 325 being equal to 20.
So, the equation is:
[tex]\[ |w - 325| = 20 \][/tex]
### Find the Highest and Lowest Weights:
3. Solve the Absolute Value Equation:
- An absolute value equation [tex]\( |x - a| = b \)[/tex] has two solutions: [tex]\( x = a + b \)[/tex] and [tex]\( x = a - b \)[/tex].
- Applying this to [tex]\( |w - 325| = 20 \)[/tex]:
[tex]\[
\begin{cases}
w - 325 = 20 & \text{or} \\
w - 325 = -20
\end{cases}
\][/tex]
4. Calculate the Solutions:
- For [tex]\( w - 325 = 20 \)[/tex]:
[tex]\[
w = 325 + 20 = 345
\][/tex]
- For [tex]\( w - 325 = -20 \)[/tex]:
[tex]\[
w = 325 - 20 = 305
\][/tex]
Therefore, the highest weight the athlete can lift is 345 pounds, and the lowest weight the athlete can lift is 305 pounds.
### Conclusion:
The equation that correctly represents the variation in the weight the athlete can lift is:
[tex]\[ |w - 325| = 20 \][/tex]
And the possible range of weights the athlete can lift is from 305 to 345 pounds.
