Let's solve the inequality step by step:

Given inequality:

$-9 + 6x \geq 39$

Step 1: Add 9 to both sides to eliminate the constant term on the left-hand side:

$-9 + 6x + 9 \geq 39 + 9$ $6x \geq 48$

Step 2: Divide both sides by 6 to isolate $x$:

$x \geq \frac{48}{6}$ $x \geq 8$

Final solution:

$x \geq 8$

Graphing the solution:

To graph this inequality on a number line:

1. Draw a number line and mark the point $x = 8$.
2. Since the inequality includes "greater than or equal to" ( $\geq$ ), put a closed circle at $x = 8$ to indicate that $x = 8$ is included in the solution.
3. Shade the line to the right of 8 to show that $x$ can take any value greater than or equal to 8.

Would you like a detailed graph of this solution?

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Here are 5 related questions to further explore this concept:

1. How would the graph change if the inequality were $x > 8$ instead of $x \geq 8$?
2. How would you solve an inequality like $5x - 12 \leq 28$?
3. What is the difference between an open circle and a closed circle on a number line graph?
4. Can inequalities have more than one solution? How would you graph that?
5. How would you solve the inequality $-3x + 7 > 25$?

Tip: Always remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality symbol.