Answer :
Sure! Let's solve this problem step by step.
### Problem Breakdown
Darcie crochets at a rate of [tex]\(\frac{1}{15}\)[/tex] of a blanket per day. She wants to donate the blankets after 60 days and plans to skip crocheting some days to volunteer. We need to figure out how many days she can skip and still complete at least one full blanket.
### Step-by-Step Solution
1. Determine Work Requirement:
- Darcie needs to complete at least 1 blanket in total by the end of the 60 days.
2. Rate of Completion per Day:
- Each day, she makes [tex]\(\frac{1}{15}\)[/tex] of a blanket.
3. Expression for Total Days Worked:
- Let [tex]\(s\)[/tex] represent the number of days she skips crocheting.
- Therefore, the number of days she actually crochets is [tex]\(60 - s\)[/tex].
4. Set Up the Inequality:
- The total amount of blanket Darcie completes in [tex]\(60 - s\)[/tex] days is [tex]\(\left(\frac{1}{15}\right) \times (60 - s)\)[/tex].
- We want this to be at least 1 full blanket:
[tex]\[
\left(\frac{1}{15}\right) \times (60 - s) \geq 1
\][/tex]
5. Solve the Inequality:
- Simplify the inequality:
[tex]\[
\frac{60 - s}{15} \geq 1
\][/tex]
- Multiply both sides by 15 to eliminate the fraction:
[tex]\[
60 - s \geq 15
\][/tex]
- Solve for [tex]\(s\)[/tex]:
[tex]\[
60 - 15 \geq s
\][/tex]
- Simplify:
[tex]\[
45 \geq s
\][/tex]
6. Conclusion:
- Darcie can skip up to 45 days and still complete one full blanket within the 60-day period.
### Graphing the Solution
To graph the solution of the inequality [tex]\(s \leq 45\)[/tex]:
- Draw a number line.
- Mark 0 on the left end and 60 on the right end (because there are 60 days in total).
- Shade the region from 0 to 45 to indicate all the possible values of [tex]\(s\)[/tex].
- Place a closed circle at 45 to indicate that [tex]\(s\)[/tex] can be 45 days at maximum.
As a result, the solution set for [tex]\(s\)[/tex], the days Darcie can skip, is [tex]\(s \leq 45\)[/tex]. This means she can skip any number of days from 0 to 45 inclusively and still meet her goal.
### Problem Breakdown
Darcie crochets at a rate of [tex]\(\frac{1}{15}\)[/tex] of a blanket per day. She wants to donate the blankets after 60 days and plans to skip crocheting some days to volunteer. We need to figure out how many days she can skip and still complete at least one full blanket.
### Step-by-Step Solution
1. Determine Work Requirement:
- Darcie needs to complete at least 1 blanket in total by the end of the 60 days.
2. Rate of Completion per Day:
- Each day, she makes [tex]\(\frac{1}{15}\)[/tex] of a blanket.
3. Expression for Total Days Worked:
- Let [tex]\(s\)[/tex] represent the number of days she skips crocheting.
- Therefore, the number of days she actually crochets is [tex]\(60 - s\)[/tex].
4. Set Up the Inequality:
- The total amount of blanket Darcie completes in [tex]\(60 - s\)[/tex] days is [tex]\(\left(\frac{1}{15}\right) \times (60 - s)\)[/tex].
- We want this to be at least 1 full blanket:
[tex]\[
\left(\frac{1}{15}\right) \times (60 - s) \geq 1
\][/tex]
5. Solve the Inequality:
- Simplify the inequality:
[tex]\[
\frac{60 - s}{15} \geq 1
\][/tex]
- Multiply both sides by 15 to eliminate the fraction:
[tex]\[
60 - s \geq 15
\][/tex]
- Solve for [tex]\(s\)[/tex]:
[tex]\[
60 - 15 \geq s
\][/tex]
- Simplify:
[tex]\[
45 \geq s
\][/tex]
6. Conclusion:
- Darcie can skip up to 45 days and still complete one full blanket within the 60-day period.
### Graphing the Solution
To graph the solution of the inequality [tex]\(s \leq 45\)[/tex]:
- Draw a number line.
- Mark 0 on the left end and 60 on the right end (because there are 60 days in total).
- Shade the region from 0 to 45 to indicate all the possible values of [tex]\(s\)[/tex].
- Place a closed circle at 45 to indicate that [tex]\(s\)[/tex] can be 45 days at maximum.
As a result, the solution set for [tex]\(s\)[/tex], the days Darcie can skip, is [tex]\(s \leq 45\)[/tex]. This means she can skip any number of days from 0 to 45 inclusively and still meet her goal.
