To solve the inequality $8x + 3 \geq -21$, let's break it down step-by-step:

Step 1: Isolate the variable term

Start by subtracting 3 from both sides of the inequality: $8x + 3 - 3 \geq -21 - 3$ $8x \geq -24$

Step 2: Solve for $x$

Now, divide both sides by 8 to solve for $x$: $x \geq -3$

Solution in Inequality Notation

The solution in inequality notation is: $x \geq -3$

Solution on the Number Line

On the number line, you would:

1. Place a filled dot at $-3$ (since it includes $-3$, represented by $\geq$).
2. Shade to the right of $-3$ to represent all numbers greater than or equal to $-3$.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What if the inequality was $8x + 3 > -21$? How would the solution differ?
2. How would you represent the solution if it was $x \leq -3$ on the number line?
3. Can you solve $8x + 3 \leq -21$ and graph the solution?
4. How do inequalities change when dividing by a negative number?
5. What if the original inequality was $-8x + 3 \geq -21$?

Tip: When solving inequalities, remember to flip the inequality sign if you divide or multiply by a negative number.