Answer :
To determine how many weeks the wrestler should lose weight to qualify for his class, we need to analyze the given conditions:
1. The wrestler needs to weigh more than 165 pounds.
2. The wrestler needs to weigh less than or equal to 185 pounds.
He currently weighs 189 pounds and is losing 0.5 pounds per week.
We need to find the number of weeks [tex]\( w \)[/tex] he needs to keep losing weight such that:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
Let's break this condition into two parts and solve each one:
### Part 1: Weight greater than 165 pounds
[tex]\[ 165 < 189 - 0.5w \][/tex]
First, solve for [tex]\( w \)[/tex]:
1. Subtract 189 from both sides:
[tex]\[ 165 - 189 < -0.5w \][/tex]
2. Simplify:
[tex]\[ -24 < -0.5w \][/tex]
3. Multiply both sides by -1 (and reverse the inequality sign):
[tex]\[ 24 > 0.5w \][/tex]
4. Divide both sides by 0.5:
[tex]\[ 48 > w \][/tex]
### Part 2: Weight less than or equal to 185 pounds
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
Now, solve for [tex]\( w \)[/tex]:
1. Subtract 189 from both sides:
[tex]\[ -0.5w \leq 185 - 189 \][/tex]
2. Simplify:
[tex]\[ -0.5w \leq -4 \][/tex]
3. Multiply both sides by -1 (and reverse the inequality sign):
[tex]\[ 0.5w \geq 4 \][/tex]
4. Divide both sides by 0.5:
[tex]\[ w \geq 8 \][/tex]
### Conclusion:
To satisfy both conditions simultaneously, the wrestler must lose weight for an interval of [tex]\( w \)[/tex] that fits both:
- [tex]\( w \geq 8 \)[/tex]
- [tex]\( w < 48 \)[/tex]
Thus, the number of weeks [tex]\( w \)[/tex] should satisfy:
[tex]\[ 8 \leq w < 48 \][/tex]
Therefore, the number of weeks he should lose weight to be in the qualifying weight range is captured by the inequality [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex], which matches:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
1. The wrestler needs to weigh more than 165 pounds.
2. The wrestler needs to weigh less than or equal to 185 pounds.
He currently weighs 189 pounds and is losing 0.5 pounds per week.
We need to find the number of weeks [tex]\( w \)[/tex] he needs to keep losing weight such that:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
Let's break this condition into two parts and solve each one:
### Part 1: Weight greater than 165 pounds
[tex]\[ 165 < 189 - 0.5w \][/tex]
First, solve for [tex]\( w \)[/tex]:
1. Subtract 189 from both sides:
[tex]\[ 165 - 189 < -0.5w \][/tex]
2. Simplify:
[tex]\[ -24 < -0.5w \][/tex]
3. Multiply both sides by -1 (and reverse the inequality sign):
[tex]\[ 24 > 0.5w \][/tex]
4. Divide both sides by 0.5:
[tex]\[ 48 > w \][/tex]
### Part 2: Weight less than or equal to 185 pounds
[tex]\[ 189 - 0.5w \leq 185 \][/tex]
Now, solve for [tex]\( w \)[/tex]:
1. Subtract 189 from both sides:
[tex]\[ -0.5w \leq 185 - 189 \][/tex]
2. Simplify:
[tex]\[ -0.5w \leq -4 \][/tex]
3. Multiply both sides by -1 (and reverse the inequality sign):
[tex]\[ 0.5w \geq 4 \][/tex]
4. Divide both sides by 0.5:
[tex]\[ w \geq 8 \][/tex]
### Conclusion:
To satisfy both conditions simultaneously, the wrestler must lose weight for an interval of [tex]\( w \)[/tex] that fits both:
- [tex]\( w \geq 8 \)[/tex]
- [tex]\( w < 48 \)[/tex]
Thus, the number of weeks [tex]\( w \)[/tex] should satisfy:
[tex]\[ 8 \leq w < 48 \][/tex]
Therefore, the number of weeks he should lose weight to be in the qualifying weight range is captured by the inequality [tex]\( 165 < 189 - 0.5w \leq 185 \)[/tex], which matches:
[tex]\[ 165 < 189 - 0.5w \leq 185 \][/tex]
