Answer :
To solve the problem, we need to determine the number of weeks the wrestler should lose weight in order to fall within the qualifying weight range, which is more than 165 pounds but less than or equal to 185 pounds.
First inequality:
[tex]\[ 165 < 189 - 0.5w \][/tex]
This inequality states that the wrestler's weight after losing weight should be more than 165 pounds. Solving for [tex]\(w\)[/tex]:
1. Subtract 165 from both sides:
[tex]\[ 189 - 0.5w > 165 \][/tex]
2. Rearrange to solve for [tex]\(w\)[/tex]:
[tex]\[ 189 - 165 > 0.5w \][/tex]
3. Simplify on the left-hand side:
[tex]\[ 24 > 0.5w \][/tex]
4. Divide both sides by 0.5 to solve for [tex]\(w\)[/tex]:
[tex]\[ w < 48 \][/tex]
So, the wrestler must lose weight for fewer than 48 weeks to ensure his weight is more than 165 pounds.
Second inequality:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
This inequality indicates that the wrestler's weight should be less than or equal to 185 pounds. Solving for [tex]\(w\)[/tex]:
1. Subtract 185 from both sides:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
2. Rearrange to solve for [tex]\(w\)[/tex]:
[tex]\[ 189 - 185 \leq 0.5w \][/tex]
3. Simplify on the left-hand side:
[tex]\[ 4 \leq 0.5w \][/tex]
4. Divide both sides by 0.5 to solve for [tex]\(w\)[/tex]:
[tex]\[ w \geq 8 \][/tex]
So, the wrestler must lose weight for at least 8 weeks to ensure his weight is not more than 185 pounds.
Conclusion:
To be within the qualifying weight range, the number of weeks [tex]\(w\)[/tex] should satisfy both conditions: [tex]\(w \geq 8\)[/tex] and [tex]\(w < 48\)[/tex]. Therefore, the correct inequality that models [tex]\(w\)[/tex], the number of weeks he should lose weight, is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
This corresponds to the solution: [tex]\(w \geq 8\)[/tex] and [tex]\(w < 48\)[/tex].
First inequality:
[tex]\[ 165 < 189 - 0.5w \][/tex]
This inequality states that the wrestler's weight after losing weight should be more than 165 pounds. Solving for [tex]\(w\)[/tex]:
1. Subtract 165 from both sides:
[tex]\[ 189 - 0.5w > 165 \][/tex]
2. Rearrange to solve for [tex]\(w\)[/tex]:
[tex]\[ 189 - 165 > 0.5w \][/tex]
3. Simplify on the left-hand side:
[tex]\[ 24 > 0.5w \][/tex]
4. Divide both sides by 0.5 to solve for [tex]\(w\)[/tex]:
[tex]\[ w < 48 \][/tex]
So, the wrestler must lose weight for fewer than 48 weeks to ensure his weight is more than 165 pounds.
Second inequality:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
This inequality indicates that the wrestler's weight should be less than or equal to 185 pounds. Solving for [tex]\(w\)[/tex]:
1. Subtract 185 from both sides:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
2. Rearrange to solve for [tex]\(w\)[/tex]:
[tex]\[ 189 - 185 \leq 0.5w \][/tex]
3. Simplify on the left-hand side:
[tex]\[ 4 \leq 0.5w \][/tex]
4. Divide both sides by 0.5 to solve for [tex]\(w\)[/tex]:
[tex]\[ w \geq 8 \][/tex]
So, the wrestler must lose weight for at least 8 weeks to ensure his weight is not more than 185 pounds.
Conclusion:
To be within the qualifying weight range, the number of weeks [tex]\(w\)[/tex] should satisfy both conditions: [tex]\(w \geq 8\)[/tex] and [tex]\(w < 48\)[/tex]. Therefore, the correct inequality that models [tex]\(w\)[/tex], the number of weeks he should lose weight, is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
This corresponds to the solution: [tex]\(w \geq 8\)[/tex] and [tex]\(w < 48\)[/tex].
