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 equation models [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]

Let the wrestler’s weight after [tex]$w$[/tex] weeks be given by

[tex]$$
\text{weight} = 189 - 0.5w.
$$[/tex]

To qualify, his weight must be more than [tex]$165$[/tex] pounds and less than or equal to [tex]$185$[/tex] pounds. This gives us the inequality

[tex]$$
165 < 189 - 0.5w \le 185.
$$[/tex]

This model correctly represents the condition that his weight must be strictly greater than [tex]$165$[/tex] pounds and at most [tex]$185$[/tex] pounds.

Next, we solve each part of the inequality to determine the range for [tex]$w$[/tex].

1. For the upper bound:

[tex]$$
189 - 0.5w \le 185.
$$[/tex]

Subtract [tex]$189$[/tex] from both sides:

[tex]$$
-0.5w \le 185 - 189 = -4.
$$[/tex]

Multiply both sides by [tex]$-2$[/tex] (remembering to reverse the inequality):

[tex]$$
w \ge 8.
$$[/tex]

2. For the lower bound:

[tex]$$
189 - 0.5w > 165.
$$[/tex]

Subtract [tex]$189$[/tex] from both sides:

[tex]$$
-0.5w > 165 - 189 = -24.
$$[/tex]

Multiply both sides by [tex]$-2$[/tex] (again reversing the inequality):

[tex]$$
w < 48.
$$[/tex]

Thus, the number of weeks must satisfy

[tex]$$
8 \le w < 48.
$$[/tex]

Among the provided models, the inequality

[tex]$$
165 < 189 - 0.5w \le 185
$$[/tex]

is the one that correctly reflects the problem's requirements. Therefore, this is the correct choice.