Sure, I'll guide you through the solution step-by-step for the inequality [tex]\(1 \geq -(q - 6) - 1\)[/tex].
### Step-by-Step Solution
1. Simplify the expression inside the parentheses:
Start with the given inequality:
[tex]\[
1 \geq -(q - 6) - 1
\][/tex]
Distribute the negative sign through the parentheses:
[tex]\[
1 \geq -q + 6 - 1
\][/tex]
2. Combine like terms:
Combine the constant terms on the right-hand side:
[tex]\[
1 \geq -q + 5
\][/tex]
3. Isolate the variable [tex]\( q \)[/tex]:
To isolate [tex]\( q \)[/tex], subtract 5 from both sides of the inequality:
[tex]\[
1 - 5 \geq -q
\][/tex]
Simplify the left-hand side:
[tex]\[
-4 \geq -q
\][/tex]
4. Solve for [tex]\( q \)[/tex]:
Since we need to solve for [tex]\( q \)[/tex], multiply both sides by -1. Keep in mind that multiplying by a negative number reverses the inequality sign:
[tex]\[
4 \leq q
\][/tex]
This can also be written as:
[tex]\[
q \geq 4
\][/tex]
### Graphing the Solution
To graph the solution [tex]\( q \geq 4 \)[/tex]:
1. Identify the Endpoint:
The inequality [tex]\( q \geq 4 \)[/tex] includes 4 itself because of the "greater than or equal to" sign ( [tex]\(\geq\)[/tex]). Thus, 4 is a closed endpoint (indicated by a solid dot).
2. Draw the Ray:
Since [tex]\( q \)[/tex] is greater than or equal to 4, you will draw a ray starting at 4 and extending indefinitely to the right.
- Place a solid dot at [tex]\( q = 4 \)[/tex].
- Draw a ray starting from this solid dot and extending to the right on the number line.
This graph visually represents all values of [tex]\( q \)[/tex] that satisfy the inequality [tex]\( q \geq 4 \)[/tex]. Here's a simple sketch of it:
```
Number Line: -∞ ... 3 4 5 6 7 ...
•-----→
```
The solid dot at 4 indicates that 4 is part of the solution (closed interval), and the arrow to the right shows that all numbers greater than 4 are also part of the solution.
