Question

You found that -4 is the boundary point and is not a solution to the inequality. How can you represent this on a number line?

r - 6 < -10

r-6=-10r=-4

-7

-6

-5

-4

-3

-2

-1

-7

-6

-5

-4

-3

-2

-1

Answer 1

To represent the boundary point and the solution to the inequality \( r - 6 < -10 \) on a number line, we first need to solve the inequality for \( r \).

1. Start with the inequality:
\[
r - 6 < -10
\]

2. Add 6 to both sides to isolate \( r \):
\[
r < -10 + 6
\]
\[
r < -4
\]

Now, this means that \( r \) can take any value less than -4. The point -4 itself is not included in the solution set (i.e., it's a boundary point) because the inequality is strict (<), not inclusive (≤).

### Representing on the Number Line

- Draw a horizontal line and mark points in the appropriate range, specifically highlighting -7, -6, -5, -4, -3, -2, -1.
- Since -4 is not included in the solution, represent -4 with an **open circle** on the number line.
- Shade to the left of -4 to indicate all values that are less than -4 are part of the solution set.

Here’s what it looks like:

```
<---o---|---|---|---|---|---|---|---|---|--->
-7 -6 -5 -4 -3 -2 -1

Shaded to the left of -4 (not including -4)
```

In summary:
- The open circle at -4 indicates that it is a boundary point and not part of the solution.
- The shading to the left indicates that all values less than -4 are included in the solution.