https://file.aiquickdraw.com/mathbot/file/172952117528851yop1bn.jpg

The problem in your image consists of solving a compound inequality and graphing the solution on a number line. The inequalities given are:

\[
1 \geq x - 1 \quad \text{and} \quad -x - 1 \geq -13
\]

Let's solve each inequality step by step.

### First Inequality:
\[
1 \geq x - 1
\]
Add \(1\) to both sides:
\[
1 + 1 \geq x
\]
\[
2 \geq x
\]
This simplifies to:
\[
x \leq 2
\]

### Second Inequality:
\[
-x - 1 \geq -13
\]
Add \(1\) to both sides:
\[
-x \geq -12
\]
Now, multiply both sides by \(-1\), remembering to reverse the inequality:
\[
x \leq 12
\]

### Final Solution:
The two inequalities we found are:
\[
x \leq 2 \quad \text{and} \quad x \leq 12
\]

Since the first inequality is more restrictive, the solution is simply:
\[
x \leq 2
\]

### Inequality Notation:
The solution in inequality notation is:
\[
x \leq 2
\]

### Graphing the Solution:
On the number line, you will mark \(x = 2\) with a filled (closed) circle because the inequality includes \(2\) (i.e., \(x\) is less than or equal to 2). Shade the number line to the left of 2, indicating all values less than or equal to 2.

Would you like further clarification on any step? Here are 5 related questions to deepen your understanding:

1. How do you solve compound inequalities with "and" versus "or" conditions?
2. What happens to an inequality when you multiply or divide by a negative number?
3. How can you verify the solution of an inequality?
4. What are the differences between open and closed circles on a number line?
5. How would the solution change if one of the inequalities were "greater than" instead of "greater than or equal to"?

**Tip**: Always check if you need to reverse the inequality when multiplying or dividing by a negative number!

Solve for x and graph the solution on the number line for the inequalities 1 ≥ x - 1 and -x - 1 ≥ -13.

This problem involves solving the compound inequalities 1 ≥ x - 1 and -x - 1 ≥ -13, followed by graphing the solution on a number line. Learn step-by-step how to isolate x and graph the solution with a closed circle at x = 2.

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