Answer :
Sure! Let's break down the problem step by step to understand which inequality models the number of weeks the wrestler should lose weight to qualify in his weight class.
### Understanding the Problem
The wrestler needs to weigh more than 165 pounds and less than or equal to 185 pounds. His current weight is 189 pounds, and he loses 0.5 pounds per week. We need to form an inequality that defines how his weight loss can fit within the qualifying range.
### Setting Up the Inequality
1. Weight after w weeks:
The formula for the wrestler's weight after losing 0.5 pounds each week can be expressed as:
[tex]\[ \text{Weight after } w \text{ weeks} = 189 - 0.5w \][/tex]
2. Qualifying Weight Range:
The qualifying weight range is defined as more than 165 pounds but less than or equal to 185 pounds:
[tex]\[ 165 < \text{Weight} \leq 185 \][/tex]
3. Substituting the Weight Formula:
Now substitute the expression for the weight after w weeks into the inequality:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
### Solving the Inequality
To find w, we solve the inequality in two parts:
Part 1:
[tex]\[ 165 < 189 - 0.5w \][/tex]
Subtract 189 from both sides:
[tex]\[ 165 - 189 < -0.5w \][/tex]
This simplifies to:
[tex]\[ -24 < -0.5w \][/tex]
Divide both sides by -0.5, remembering to flip the inequality because you're dividing by a negative number:
[tex]\[ 48 > w \][/tex]
Which translates to:
[tex]\[ w < 48 \][/tex]
Part 2:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
Subtract 189 from both sides:
[tex]\[ -0.5w \leq 185 - 189 \][/tex]
This simplifies to:
[tex]\[ -0.5w \leq -4 \][/tex]
Divide both sides by -0.5, flipping the inequality:
[tex]\[ w \geq 8 \][/tex]
### Combining the Results
Combining these two results gives us the range for w:
[tex]\[ 8 \leq w < 48 \][/tex]
This means the wrestler has to lose weight for at least 8 weeks and no more than 47 weeks (since w is strictly less than 48) to qualify in his weight class. This matches our options, confirming that the correct inequality is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
### Understanding the Problem
The wrestler needs to weigh more than 165 pounds and less than or equal to 185 pounds. His current weight is 189 pounds, and he loses 0.5 pounds per week. We need to form an inequality that defines how his weight loss can fit within the qualifying range.
### Setting Up the Inequality
1. Weight after w weeks:
The formula for the wrestler's weight after losing 0.5 pounds each week can be expressed as:
[tex]\[ \text{Weight after } w \text{ weeks} = 189 - 0.5w \][/tex]
2. Qualifying Weight Range:
The qualifying weight range is defined as more than 165 pounds but less than or equal to 185 pounds:
[tex]\[ 165 < \text{Weight} \leq 185 \][/tex]
3. Substituting the Weight Formula:
Now substitute the expression for the weight after w weeks into the inequality:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
### Solving the Inequality
To find w, we solve the inequality in two parts:
Part 1:
[tex]\[ 165 < 189 - 0.5w \][/tex]
Subtract 189 from both sides:
[tex]\[ 165 - 189 < -0.5w \][/tex]
This simplifies to:
[tex]\[ -24 < -0.5w \][/tex]
Divide both sides by -0.5, remembering to flip the inequality because you're dividing by a negative number:
[tex]\[ 48 > w \][/tex]
Which translates to:
[tex]\[ w < 48 \][/tex]
Part 2:
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
Subtract 189 from both sides:
[tex]\[ -0.5w \leq 185 - 189 \][/tex]
This simplifies to:
[tex]\[ -0.5w \leq -4 \][/tex]
Divide both sides by -0.5, flipping the inequality:
[tex]\[ w \geq 8 \][/tex]
### Combining the Results
Combining these two results gives us the range for w:
[tex]\[ 8 \leq w < 48 \][/tex]
This means the wrestler has to lose weight for at least 8 weeks and no more than 47 weeks (since w is strictly less than 48) to qualify in his weight class. This matches our options, confirming that the correct inequality is:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
