Answer :
To solve this problem, we need to find out how many weeks it will take for the wrestler to be within his qualifying weight class, which requires him to weigh more than 165 pounds and less than or equal to 185 pounds. His current weight is 189 pounds, and he loses 0.5 pounds every week.
We can set up the inequality to model this situation as:
1. The weight must be greater than 165 pounds:
[tex]\( 165 < 189 - 0.5w \)[/tex]
2. The weight must be less than or equal to 185 pounds:
[tex]\( 189 - 0.5w \leq 185 \)[/tex]
Let's solve each part of this compound inequality:
Step 1: Solve [tex]\( 165 < 189 - 0.5w \)[/tex]
- Subtract 189 from both sides:
[tex]\( 165 - 189 < -0.5w \)[/tex]
[tex]\(-24 < -0.5w \)[/tex]
- Divide by [tex]\(-0.5\)[/tex], and remember to flip the inequality sign:
[tex]\( w > \frac{-24}{-0.5} \)[/tex]
[tex]\( w > 48 \)[/tex]
This means the wrestler needs more than 48 weeks to weigh more than 165 pounds.
Step 2: Solve [tex]\( 189 - 0.5w \leq 185 \)[/tex]
- Subtract 189 from both sides:
[tex]\( 189 - 189 - 0.5w \leq 185 - 189 \)[/tex]
[tex]\(-0.5w \leq -4 \)[/tex]
- Divide by [tex]\(-0.5\)[/tex], remembering to flip the inequality sign:
[tex]\( w \geq \frac{-4}{-0.5} \)[/tex]
[tex]\( w \leq 8 \)[/tex]
This means the wrestler should not take more than 8 weeks to be within the upper limit of 185 pounds.
Conclusion:
Combining these results, to be in the qualifying weight range, the solution indicates:
The wrestler must lose weight for more than 48 weeks to weigh more than 165 pounds but also within 8 weeks of the weight being reduced to 185 pounds or less. The values calculated, as provided, indicate that this model [tex]\(165 < 189 - 0.5w \leq 185\)[/tex] correctly captures the wrestler's qualifying conditions.
We can set up the inequality to model this situation as:
1. The weight must be greater than 165 pounds:
[tex]\( 165 < 189 - 0.5w \)[/tex]
2. The weight must be less than or equal to 185 pounds:
[tex]\( 189 - 0.5w \leq 185 \)[/tex]
Let's solve each part of this compound inequality:
Step 1: Solve [tex]\( 165 < 189 - 0.5w \)[/tex]
- Subtract 189 from both sides:
[tex]\( 165 - 189 < -0.5w \)[/tex]
[tex]\(-24 < -0.5w \)[/tex]
- Divide by [tex]\(-0.5\)[/tex], and remember to flip the inequality sign:
[tex]\( w > \frac{-24}{-0.5} \)[/tex]
[tex]\( w > 48 \)[/tex]
This means the wrestler needs more than 48 weeks to weigh more than 165 pounds.
Step 2: Solve [tex]\( 189 - 0.5w \leq 185 \)[/tex]
- Subtract 189 from both sides:
[tex]\( 189 - 189 - 0.5w \leq 185 - 189 \)[/tex]
[tex]\(-0.5w \leq -4 \)[/tex]
- Divide by [tex]\(-0.5\)[/tex], remembering to flip the inequality sign:
[tex]\( w \geq \frac{-4}{-0.5} \)[/tex]
[tex]\( w \leq 8 \)[/tex]
This means the wrestler should not take more than 8 weeks to be within the upper limit of 185 pounds.
Conclusion:
Combining these results, to be in the qualifying weight range, the solution indicates:
The wrestler must lose weight for more than 48 weeks to weigh more than 165 pounds but also within 8 weeks of the weight being reduced to 185 pounds or less. The values calculated, as provided, indicate that this model [tex]\(165 < 189 - 0.5w \leq 185\)[/tex] correctly captures the wrestler's qualifying conditions.
