Answer :
Of course! Let's carefully work through the question step-by-step.
The wrestler currently weighs 189 pounds and is losing weight at a rate of 0.5 pounds per week. We need to find out for how many weeks he should lose weight in order to weigh more than 165 pounds but less than or equal to 185 pounds.
1. Set Up the Inequality:
- We want his weight to be within [tex]\(165 < \text{weight} \leq 185\)[/tex].
- Let's denote the number of weeks as [tex]\( w \)[/tex].
- His weight after [tex]\( w \)[/tex] weeks is given by [tex]\( 189 - 0.5w \)[/tex].
2. Formulate the Inequalities:
- Combine this information into one compound inequality:
[tex]\[
165 < 189 - 0.5w \leq 185
\][/tex]
3. Solve the Inequalities:
- Let's split this compound inequality into two separate parts and solve each part.
- First Inequality: [tex]\( 165 < 189 - 0.5w \)[/tex]
[tex]\[
165 < 189 - 0.5w
\][/tex]
[tex]\[
165 - 189 < -0.5w
\][/tex]
[tex]\[
-24 < -0.5w
\][/tex]
Divide both sides by -0.5 (remember to reverse the inequality sign):
[tex]\[
48 > w
\][/tex]
Simplified, we get:
[tex]\[
w < 48
\][/tex]
- Second Inequality: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
[tex]\[
189 - 0.5w \leq 185
\][/tex]
[tex]\[
189 - 185 \leq 0.5w
\][/tex]
[tex]\[
4 \leq 0.5w
\][/tex]
Divide both sides by 0.5:
[tex]\[
8 \leq w
\][/tex]
4. Combine the Results:
- From the solutions of both inequalities, we get:
[tex]\[
8 \leq w < 48
\][/tex]
So, the wrestler needs to lose weight for a period of time within the range of 8 to 48 weeks to qualify in his weight class.
Thus, the correct option is:
[tex]\[
165 < 189 - 0.5w \leq 185
\][/tex]
The wrestler currently weighs 189 pounds and is losing weight at a rate of 0.5 pounds per week. We need to find out for how many weeks he should lose weight in order to weigh more than 165 pounds but less than or equal to 185 pounds.
1. Set Up the Inequality:
- We want his weight to be within [tex]\(165 < \text{weight} \leq 185\)[/tex].
- Let's denote the number of weeks as [tex]\( w \)[/tex].
- His weight after [tex]\( w \)[/tex] weeks is given by [tex]\( 189 - 0.5w \)[/tex].
2. Formulate the Inequalities:
- Combine this information into one compound inequality:
[tex]\[
165 < 189 - 0.5w \leq 185
\][/tex]
3. Solve the Inequalities:
- Let's split this compound inequality into two separate parts and solve each part.
- First Inequality: [tex]\( 165 < 189 - 0.5w \)[/tex]
[tex]\[
165 < 189 - 0.5w
\][/tex]
[tex]\[
165 - 189 < -0.5w
\][/tex]
[tex]\[
-24 < -0.5w
\][/tex]
Divide both sides by -0.5 (remember to reverse the inequality sign):
[tex]\[
48 > w
\][/tex]
Simplified, we get:
[tex]\[
w < 48
\][/tex]
- Second Inequality: [tex]\( 189 - 0.5w \leq 185 \)[/tex]
[tex]\[
189 - 0.5w \leq 185
\][/tex]
[tex]\[
189 - 185 \leq 0.5w
\][/tex]
[tex]\[
4 \leq 0.5w
\][/tex]
Divide both sides by 0.5:
[tex]\[
8 \leq w
\][/tex]
4. Combine the Results:
- From the solutions of both inequalities, we get:
[tex]\[
8 \leq w < 48
\][/tex]
So, the wrestler needs to lose weight for a period of time within the range of 8 to 48 weeks to qualify in his weight class.
Thus, the correct option is:
[tex]\[
165 < 189 - 0.5w \leq 185
\][/tex]
