The inequality given is:

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

Let's solve each inequality separately:

Solving the First Inequality:

$1 \geq x - 9$ To isolate $x$, add 9 to both sides: $1 + 9 \geq x$ $10 \geq x \quad \Rightarrow \quad x \leq 10$

Solving the Second Inequality:

$x - 9 > 3$ Again, add 9 to both sides to isolate $x$: $x > 3 + 9$ $x > 12$

Combined Solution

Since the solution is given in the form of an "or" statement, we include values that satisfy either inequality.

1. From the first inequality $x \leq 10$.
2. From the second inequality $x > 12$.

The combined solution is: $x \leq 10 \quad \text{or} \quad x > 12$

Graphing on the Number Line

On the number line:

• Draw a closed circle at 10 and shade all values to the left to represent $x \leq 10$.
• Draw an open circle at 12 and shade all values to the right to represent $x > 12$.

This shows that the solution includes all values $x \leq 10$ and $x > 12$, with a gap between 10 and 12.

Would you like further details on graphing this or have any questions?

Here are some related questions:

1. What would the solution look like if it were $1 \geq x - 9$ and $x - 9 > 3$?
2. How would we solve $1 \leq x - 9$ or $x - 9 < 3$?
3. Can we rewrite the solution set using interval notation?
4. What is the difference between “and” and “or” in inequalities?
5. How can we verify the solution on a number line?

Tip: For compound inequalities, always check if it’s an "and" (intersection) or "or" (union) problem, as it affects the solution.