Sure, let's address the problem step by step:
### Step 1: Define Darcie's Crocheting Rate
Darcie can crochet [tex]\(\frac{1}{15}\)[/tex] blankets per day.
### Step 2: Total Days Available
Darcie has a total of 60 days available to crochet.
### Step 3: Minimum Number of Blankets
Darcie wants to crochet at least 3 blankets.
### Step 4: Calculate the Days Required to Crochet 3 Blankets
To determine how many days, [tex]\(d\)[/tex], she needs to crochet 3 blankets, we can use the formula:
[tex]\[
\text{Number of days} = \frac{\text{Number of blankets}}{\text{Crocheting rate}}
\][/tex]
For Darcie:
[tex]\[
d = \frac{3}{\frac{1}{15}} = 3 \times 15 = 45 \text{ days}
\][/tex]
### Step 5: Calculate the Number of Days She Can Skip
Since Darcie has 60 days available and she needs 45 days to crochet the 3 blankets, she can skip:
[tex]\[
s = 60 - 45 = 15 \text{ days}
\][/tex]
### Step 6: Write the Inequality
To generalize this, we need the inequality that shows the number of days, [tex]\(s\)[/tex], Darcie can skip and still meet her goal:
[tex]\[
s \leq 60 - 45
\][/tex]
Simplifying, we have:
[tex]\[
s \leq 15
\][/tex]
### Graph the Solution Set
To graph the solution set of the inequality [tex]\(s \leq 15\)[/tex]:
1. Draw a number line.
2. Mark the point [tex]\(15\)[/tex] on the number line.
3. Include all the numbers to the left of [tex]\(15\)[/tex] (because [tex]\(s \leq 15\)[/tex]).
4. Use a closed circle at [tex]\(15\)[/tex] to indicate that [tex]\(15\)[/tex] is included in the solution set.
```
|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
<-----------------------[===========allowed days Darcie can skip===================]----->
```
Here, [tex]\(s\)[/tex] represents any number of days from [tex]\(0\)[/tex] to [tex]\(15\)[/tex] inclusively that Darcie can skip crocheting and still manage to crochet at least 3 blankets within 60 days.
