Question

All is collecting signatures for a petition.
• He currently has 520
signatures.
​• He has 6
more weeks to collect the remaining signatures he needs.
​• He needs a total of at least
1000 signatures before he can submit the petition.
​
Ali wants to collect the same number of signatures each week.
​
​Which number line represents all possible numbers of signatures Ali could collect in each of the remaining weeks so that he will have enough signatures to submit the petition?

Answer 1

To determine how many signatures Ali needs to collect each week for the next 6 weeks, we can follow these steps:

1. **Calculate Total Signatures Needed**:
Ali needs a total of 1000 signatures. He currently has 520 signatures.
\[
\text{Signatures needed} = 1000 - 520 = 480
\]

2. **Determine Weekly Signature Collection**:
The number of signatures Ali needs to collect each week over the next 6 weeks can be expressed as \( x \). Therefore, over 6 weeks, he will collect \( 6x \) signatures. We need to satisfy the condition that:
\[
6x \geq 480
\]

3. **Solving the Inequality**:
To find the minimum number of signatures per week, we divide both sides of the inequality by 6:
\[
x \geq \frac{480}{6} = 80
\]
This means Ali needs to collect at least 80 signatures each week.

4. **Number Line Representation**:
The solution \( x \geq 80 \) can be represented on a number line by indicating all values starting from 80 and extending to the right towards infinity.

Thus, the number line representing all possible numbers of signatures Ali could collect in each of the remaining weeks is marked at 80 with a closed circle (indicating that 80 is included) and an arrow extending to the right, indicating all numbers greater than or equal to 80.

The number line will look like this:

```
<---|----|----|----|----|----|----|----|----->
0 20 40 60 80 100 120 ...
●
```

Where the dot (●) on 80 indicates that Ali can collect 80 signatures or more each week to meet his goal.