We are given the inequality:

$1 \geq x - 9 \quad \text{or} \quad x - 9 > 3$

We need to solve for $x$ in both inequalities and graph the solutions on the number line.

First inequality: $1 \geq x - 9$

1. Add 9 to both sides of the inequality to isolate $x$:

$1 + 9 \geq x \quad \Rightarrow \quad 10 \geq x$

This simplifies to:

$x \leq 10$

Second inequality: $x - 9 > 3$

1. Add 9 to both sides of the inequality to isolate $x$:

$x - 9 + 9 > 3 + 9 \quad \Rightarrow \quad x > 12$

Combining the solutions

We have two inequalities:

• $x \leq 10$ (from the first inequality)
• $x > 12$ (from the second inequality)

Since we are using "or" in the original problem, the solution is the union of the two inequalities. So, the solution is:

$x \leq 10 \quad \text{or} \quad x > 12$

Graphing the solution on the number line

• For $x \leq 10$, draw a solid circle at 10 and shade everything to the left.
• For $x > 12$, draw an open circle at 12 and shade everything to the right.

Here's how the number line will look:

<====●------------------○====>
      10               12

• The solid circle at 10 indicates that 10 is included in the solution (since $x \leq 10$).
• The open circle at 12 indicates that 12 is not included in the solution (since $x > 12$).

Would you like more details or clarification on any part of this solution?

Related Questions:

1. How would the graph change if the inequality used "and" instead of "or"?
2. What happens if the original inequality were $x - 9 \geq 3$?
3. Can you solve and graph the inequality $x + 5 \geq 3$ or $x - 4 > 2$?
4. How would the solution change if the inequality were $1 \leq x - 9$ and $x - 9 \geq 3$?
5. What is the significance of open and closed circles on a number line?

Tip: When solving inequalities, always pay attention to whether the inequality is strict (using > or <) or non-strict (using ≥ or ≤), as this determines whether to use open or closed circles on the number line.