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Answer :
"The correct model to represent the number of weeks, w, that the wrestler should lose weight to be in the qualifying weight range is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
To understand why this is the correct model, let's break down the problem and the inequalities:
1. The wrestler currently weighs 189 pounds and needs to weigh less than or equal to 185 pounds to qualify. This means he must lose weight.
2. The wrestler is losing 0.5 of a pound per week. Therefore, each week that passes, his weight decreases by 0.5 pounds.
3. We want to find out how many weeks, w, it will take for his weight to be within the qualifying range, which is more than 165 pounds but less than or equal to 185 pounds.
4. The weight loss over w weeks is represented by the term [tex]\( 0.5w \).[/tex]So, the wrestler's weight after w weeks will be [tex]\( 189 - 0.5w \).[/tex]
5. For the wrestler to qualify, his weight after losing weight for w weeks must satisfy two conditions:
- It must be greater than 165 pounds: [tex]\( 189 - 0.5w > 165 \)[/tex]
- It must be less than or equal to 185 pounds: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
6. We can solve the first inequality to find the minimum number of weeks required for the wrestler to weigh more than 165 pounds:
[tex]\[ 189 - 0.5w > 165 \][/tex]
[tex]\[ -0.5w > 165 - 189 \][/tex]
[tex]\[ -0.5w > -24 \][/tex]
[tex]\[ w < 48 \][/tex]
Since w must be a whole number of weeks, the minimum number of weeks is 47 (since 48 weeks would put him at or below 165 pounds, which is outside the qualifying range).
7. We can solve the second inequality to find the maximum number of weeks required for the wrestler to weigh less than or equal to 185 pounds:
\[ 189 - 0.5w \leq 185 \]
\[ -0.5w \leq 185 - 189 \]
\[ -0.5w \leq -4 \]
\[ w \geq 8 \]
The wrestler needs to lose weight for at least 8 weeks to be at or below 185 pounds.
8. Combining both conditions, the wrestler needs to lose weight for a number of weeks that is greater than or equal to 8 but less than 48. This can be represented as:
[tex]\[ 8 \leq w < 48 \][/tex]
Therefore, the correct model is [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex], which translates to the wrestler needing to lose weight for 8 to 47 weeks to qualify for his weight class."
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