Answer :
To solve the problem of finding the correct model to represent the number of weeks [tex]\( w \)[/tex] the wrestler should lose weight to be in the qualifying weight range, let's analyze each inequality provided:
The wrestler needs to weigh more than 165 pounds but less than or equal to 185 pounds. He currently weighs 189 pounds and loses 0.5 pounds per week. We need to find out which inequality correctly reflects this requirement.
1. Inequality Model 1: [tex]\( 165 \leq 189 - 0.5w < 185 \)[/tex]
- Breaking it down:
- [tex]\( 165 \leq 189 - 0.5w \)[/tex] implies [tex]\( 0.5w \leq 189 - 165 \)[/tex], so [tex]\( 0.5w \leq 24 \)[/tex]. This gives us [tex]\( w \leq 48 \)[/tex].
- [tex]\( 189 - 0.5w < 185 \)[/tex] implies [tex]\( 0.5w > 189 - 185 \)[/tex], so [tex]\( 0.5w > 4 \)[/tex]. This leads to [tex]\( w > 8 \)[/tex].
- Combining these gives: [tex]\( 8 < w \leq 48 \)[/tex]
2. Inequality Model 2: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
- Breaking it down:
- [tex]\( 165 < 189 - 0.5w \)[/tex] implies [tex]\( 0.5w < 189 - 165 \)[/tex], so [tex]\( 0.5w < 24 \)[/tex]. This gives [tex]\( w < 48 \)[/tex].
- [tex]\( 189 - 0.5w \leq 185 \)[/tex] implies [tex]\( 0.5w \geq 189 - 185 \)[/tex], so [tex]\( 0.5w \geq 4 \)[/tex]. This leads to [tex]\( w \geq 8 \)[/tex].
- Combining these gives: [tex]\( 8 \leq w < 48 \)[/tex]
3. Inequality Model 3: [tex]\( 165 > 189 - 0.5w \)[/tex] or [tex]\( 185 \leq 189 - 0.5w \)[/tex]
- This doesn't reflect the requirement of being within 165 and 185 pounds. This looks for weights outside the range.
4. Inequality Model 4: [tex]\( 165 \geq 189 - 0.5w \)[/tex] or [tex]\( 185 < 189 - 0.5w \)[/tex]
- Similarly, this inequality also doesn't reflect the requirement as it suggests being outside the desired range.
Considering the requirement that the weight must be more than 165 and less than or equal to 185 pounds, Inequality Model 2: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex] is the correct one. This model ensures that his weight falls strictly between 165 and 185 pounds, inclusive on the upper end. Therefore, the wrestler should lose weight for [tex]\( 8 \leq w < 48 \)[/tex] weeks to qualify.
The wrestler needs to weigh more than 165 pounds but less than or equal to 185 pounds. He currently weighs 189 pounds and loses 0.5 pounds per week. We need to find out which inequality correctly reflects this requirement.
1. Inequality Model 1: [tex]\( 165 \leq 189 - 0.5w < 185 \)[/tex]
- Breaking it down:
- [tex]\( 165 \leq 189 - 0.5w \)[/tex] implies [tex]\( 0.5w \leq 189 - 165 \)[/tex], so [tex]\( 0.5w \leq 24 \)[/tex]. This gives us [tex]\( w \leq 48 \)[/tex].
- [tex]\( 189 - 0.5w < 185 \)[/tex] implies [tex]\( 0.5w > 189 - 185 \)[/tex], so [tex]\( 0.5w > 4 \)[/tex]. This leads to [tex]\( w > 8 \)[/tex].
- Combining these gives: [tex]\( 8 < w \leq 48 \)[/tex]
2. Inequality Model 2: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
- Breaking it down:
- [tex]\( 165 < 189 - 0.5w \)[/tex] implies [tex]\( 0.5w < 189 - 165 \)[/tex], so [tex]\( 0.5w < 24 \)[/tex]. This gives [tex]\( w < 48 \)[/tex].
- [tex]\( 189 - 0.5w \leq 185 \)[/tex] implies [tex]\( 0.5w \geq 189 - 185 \)[/tex], so [tex]\( 0.5w \geq 4 \)[/tex]. This leads to [tex]\( w \geq 8 \)[/tex].
- Combining these gives: [tex]\( 8 \leq w < 48 \)[/tex]
3. Inequality Model 3: [tex]\( 165 > 189 - 0.5w \)[/tex] or [tex]\( 185 \leq 189 - 0.5w \)[/tex]
- This doesn't reflect the requirement of being within 165 and 185 pounds. This looks for weights outside the range.
4. Inequality Model 4: [tex]\( 165 \geq 189 - 0.5w \)[/tex] or [tex]\( 185 < 189 - 0.5w \)[/tex]
- Similarly, this inequality also doesn't reflect the requirement as it suggests being outside the desired range.
Considering the requirement that the weight must be more than 165 and less than or equal to 185 pounds, Inequality Model 2: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex] is the correct one. This model ensures that his weight falls strictly between 165 and 185 pounds, inclusive on the upper end. Therefore, the wrestler should lose weight for [tex]\( 8 \leq w < 48 \)[/tex] weeks to qualify.
