Answer :
To determine the correct inequality model for the problem, we need to establish the condition where the wrestler's weight fits within the given weight class, specifically between more than 165 pounds and less than or equal to 185 pounds.
Let's break this down:
1. Current Situation:
- The wrestler weighs 189 pounds currently.
2. Weight Loss Rate:
- He is losing 0.5 pounds per week.
3. Required Weight Range:
- The wrestler's weight needs to be greater than 165 pounds but less than or equal to 185 pounds.
4. Set Up the Inequality:
- Use the formula: [tex]\( \text{current weight} - \text{(weight loss per week)} \times w \)[/tex]
- Plug in values we know: [tex]\( 189 - 0.5w \)[/tex]
5. Establish the Conditions:
- For the lower bound: The weight must be more than 165 pounds.
- [tex]\( 165 < 189 - 0.5w \)[/tex]
- For the upper bound: The weight must be less than or equal to 185 pounds.
- [tex]\( 189 - 0.5w \leq 185 \)[/tex]
6. Combine the Inequalities:
- Here we combine both conditions to ensure the wrestler falls within the qualifying range:
- [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
Thus, the correct inequality modeling the number of weeks he must lose weight to qualify is:
165 < 189 - 0.5w ≤ 185
This inequality ensures that the wrestler's weight after losing weight for [tex]\( w \)[/tex] weeks is within the specified range required for qualification.
Let's break this down:
1. Current Situation:
- The wrestler weighs 189 pounds currently.
2. Weight Loss Rate:
- He is losing 0.5 pounds per week.
3. Required Weight Range:
- The wrestler's weight needs to be greater than 165 pounds but less than or equal to 185 pounds.
4. Set Up the Inequality:
- Use the formula: [tex]\( \text{current weight} - \text{(weight loss per week)} \times w \)[/tex]
- Plug in values we know: [tex]\( 189 - 0.5w \)[/tex]
5. Establish the Conditions:
- For the lower bound: The weight must be more than 165 pounds.
- [tex]\( 165 < 189 - 0.5w \)[/tex]
- For the upper bound: The weight must be less than or equal to 185 pounds.
- [tex]\( 189 - 0.5w \leq 185 \)[/tex]
6. Combine the Inequalities:
- Here we combine both conditions to ensure the wrestler falls within the qualifying range:
- [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
Thus, the correct inequality modeling the number of weeks he must lose weight to qualify is:
165 < 189 - 0.5w ≤ 185
This inequality ensures that the wrestler's weight after losing weight for [tex]\( w \)[/tex] weeks is within the specified range required for qualification.
