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Answer :
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.
[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.
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