Answer :
To solve this problem, we need to determine the number of weeks (w) the wrestler should lose weight to be within the qualifying weight range, which is more than 165 pounds but less than or equal to 185 pounds. Here's how to do it step-by-step:
1. Identify Current Weight and Weight Loss Rate:
- Current weight: 189 pounds
- Weight loss rate: 0.5 pounds per week
2. Set Up the Inequalities:
- The wrestler needs to weigh more than 165 pounds: [tex]\( 165 < 189 - 0.5w \)[/tex]
- The wrestler's weight must be less than or equal to 185 pounds: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
3. Solve Each Inequality:
- First Inequality: [tex]\( 165 < 189 - 0.5w \)[/tex]
- Start by subtracting 189 from both sides: [tex]\( 165 - 189 < -0.5w \)[/tex]
- Simplify: [tex]\( -24 < -0.5w \)[/tex]
- Divide by -0.5, remembering to reverse the inequality sign: [tex]\( w < 48 \)[/tex]
- Second Inequality: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
- Subtract 189 from both sides: [tex]\( -0.5w \leq 185 - 189 \)[/tex]
- Simplify: [tex]\( -0.5w \leq -4 \)[/tex]
- Divide by -0.5, again reversing the inequality sign: [tex]\( w \geq 8 \)[/tex]
4. Combine the Results:
- The combination of these inequalities is: [tex]\( 8 \leq w < 48 \)[/tex]
5. Identify the Model:
- The correct model that represents this situation is: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
So, for the wrestler to be in the qualifying weight range, he should lose weight for a minimum of 8 weeks and less than 48 weeks. The correct model that fits this scenario is [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex].
1. Identify Current Weight and Weight Loss Rate:
- Current weight: 189 pounds
- Weight loss rate: 0.5 pounds per week
2. Set Up the Inequalities:
- The wrestler needs to weigh more than 165 pounds: [tex]\( 165 < 189 - 0.5w \)[/tex]
- The wrestler's weight must be less than or equal to 185 pounds: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
3. Solve Each Inequality:
- First Inequality: [tex]\( 165 < 189 - 0.5w \)[/tex]
- Start by subtracting 189 from both sides: [tex]\( 165 - 189 < -0.5w \)[/tex]
- Simplify: [tex]\( -24 < -0.5w \)[/tex]
- Divide by -0.5, remembering to reverse the inequality sign: [tex]\( w < 48 \)[/tex]
- Second Inequality: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
- Subtract 189 from both sides: [tex]\( -0.5w \leq 185 - 189 \)[/tex]
- Simplify: [tex]\( -0.5w \leq -4 \)[/tex]
- Divide by -0.5, again reversing the inequality sign: [tex]\( w \geq 8 \)[/tex]
4. Combine the Results:
- The combination of these inequalities is: [tex]\( 8 \leq w < 48 \)[/tex]
5. Identify the Model:
- The correct model that represents this situation is: [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex]
So, for the wrestler to be in the qualifying weight range, he should lose weight for a minimum of 8 weeks and less than 48 weeks. The correct model that fits this scenario is [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex].
