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For a wrestler to qualify in his weight class, he needs to weigh more than 165 pounds but less than or equal to 185 pounds. He currently weighs 189 pounds and is losing 0.5 of a pound per week.

Which model represents [tex]w[/tex], the number of weeks he should lose weight to be in the qualifying weight range?

A. [tex]165 \leq 189 - 0.5w \textless 185[/tex]

B. [tex]165 \textless 189 - 0.5w \leq 185[/tex]

C. [tex]165 \textgreater 189 - 0.5w[/tex] or [tex]185 \leq 189 - 0.5w[/tex]

D. [tex]165 \geq 189 - 0.5w[/tex] or [tex]185 \textless 189 - 0.5w[/tex]

Answer :

To determine how many weeks the wrestler should lose weight to qualify for his weight class, we start with his current situation and the requirements:

- Current weight: 189 pounds
- Qualifying weight range: More than 165 pounds but less than or equal to 185 pounds
- Weight loss per week: 0.5 pounds

We are looking for how long it will take for his weight to be within this range.

1. Determine when the wrestler will weigh more than 165 pounds:

Since he is losing 0.5 pounds per week, we can express his weight after [tex]\( w \)[/tex] weeks as:
[tex]\[
\text{Weight after } w \text{ weeks} = 189 - 0.5w
\][/tex]

To find when his weight will be more than 165 pounds, we set up the inequality:
[tex]\[
189 - 0.5w > 165
\][/tex]

Solving this inequality:
[tex]\[
189 - 165 > 0.5w
\][/tex]
[tex]\[
24 > 0.5w
\][/tex]
[tex]\[
\frac{24}{0.5} > w
\][/tex]
[tex]\[
48 > w
\][/tex]

This means it will take less than 48 weeks for his weight to be more than 165 pounds.

2. Determine when the wrestler will weigh less than or equal to 185 pounds:

Using the same expression for his weight after [tex]\( w \)[/tex] weeks:
[tex]\[
189 - 0.5w \leq 185
\][/tex]

Solving this inequality:
[tex]\[
189 - 185 \leq 0.5w
\][/tex]
[tex]\[
4 \leq 0.5w
\][/tex]
[tex]\[
\frac{4}{0.5} \leq w
\][/tex]
[tex]\[
8 \leq w
\][/tex]

This means it will take at least 8 weeks for his weight to be less than or equal to 185 pounds.

Combining both results, the wrestler should lose weight for a period such that [tex]\( 8 \leq w < 48 \)[/tex] weeks to be in the qualifying weight range of more than 165 pounds but less than or equal to 185 pounds.

The inequality description that models this situation is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]

So the correct model among the options given is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]

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