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Answer :
To determine the number of weeks the wrestler should lose weight to be in the qualifying weight range, we need to ensure his weight is more than 165 pounds but less than or equal to 185 pounds. He currently weighs 189 pounds and loses 0.5 pound per week.
We want to find out when:
1. His weight will be more than 165 pounds.
2. His weight will be less than or equal to 185 pounds.
Let's represent his weight after [tex]\( w \)[/tex] weeks of losing weight as [tex]\( 189 - 0.5w \)[/tex].
### Step 1: Solve for when his weight is more than 165 pounds.
We set up the inequality:
[tex]\[ 189 - 0.5w > 165 \][/tex]
To solve for [tex]\( w \)[/tex]:
- Subtract 165 from 189:
[tex]\[ 189 - 165 = 24 \][/tex]
- This gives:
[tex]\[ 24 > 0.5w \][/tex]
- Divide both sides by 0.5 to solve for [tex]\( w \)[/tex]:
[tex]\[ w < \frac{24}{0.5} \][/tex]
[tex]\[ w < 48 \][/tex]
### Step 2: Solve for when his weight is less than or equal to 185 pounds.
Set up the inequality:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
To solve for [tex]\( w \)[/tex]:
- Subtract 185 from 189:
[tex]\[ 189 - 185 = 4 \][/tex]
- This gives:
[tex]\[ 4 \leq 0.5w \][/tex]
- Divide both sides by 0.5:
[tex]\[ w \geq \frac{4}{0.5} \][/tex]
[tex]\[ w \geq 8 \][/tex]
### Conclusion:
The wrestler should lose weight for a period of weeks [tex]\( w \)[/tex] such that [tex]\( 8 \leq w < 48 \)[/tex].
From the given options, the one that fits this requirement is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
So, the correct model for [tex]\( w \)[/tex] is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
We want to find out when:
1. His weight will be more than 165 pounds.
2. His weight will be less than or equal to 185 pounds.
Let's represent his weight after [tex]\( w \)[/tex] weeks of losing weight as [tex]\( 189 - 0.5w \)[/tex].
### Step 1: Solve for when his weight is more than 165 pounds.
We set up the inequality:
[tex]\[ 189 - 0.5w > 165 \][/tex]
To solve for [tex]\( w \)[/tex]:
- Subtract 165 from 189:
[tex]\[ 189 - 165 = 24 \][/tex]
- This gives:
[tex]\[ 24 > 0.5w \][/tex]
- Divide both sides by 0.5 to solve for [tex]\( w \)[/tex]:
[tex]\[ w < \frac{24}{0.5} \][/tex]
[tex]\[ w < 48 \][/tex]
### Step 2: Solve for when his weight is less than or equal to 185 pounds.
Set up the inequality:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
To solve for [tex]\( w \)[/tex]:
- Subtract 185 from 189:
[tex]\[ 189 - 185 = 4 \][/tex]
- This gives:
[tex]\[ 4 \leq 0.5w \][/tex]
- Divide both sides by 0.5:
[tex]\[ w \geq \frac{4}{0.5} \][/tex]
[tex]\[ w \geq 8 \][/tex]
### Conclusion:
The wrestler should lose weight for a period of weeks [tex]\( w \)[/tex] such that [tex]\( 8 \leq w < 48 \)[/tex].
From the given options, the one that fits this requirement is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
So, the correct model for [tex]\( w \)[/tex] is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
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