The problem in the image is a compound inequality:

$2x + 7 < 9 \quad \text{and} \quad 2x + 7 < 21$

Let's solve each inequality step by step.

1. Solve the first inequality:

$2x + 7 < 9$ Subtract 7 from both sides: $2x < 2$ Now divide by 2: $x < 1$

2. Solve the second inequality:

$2x + 7 < 21$ Subtract 7 from both sides: $2x < 14$ Now divide by 2: $x < 7$

3. Combine the solutions:

The solution to the compound inequality is the intersection of the two individual inequalities. So, we take the stricter condition between $x < 1$ and $x < 7$.

Thus, the solution is: $x < 1$

4. Inequality notation:

The final inequality notation is: $x < 1$

Would you like further details on how to graph this on a number line?

Related Questions:

1. What would the solution be if we had an "or" instead of "and" in the compound inequality?
2. How do you represent the solution of compound inequalities graphically on a number line?
3. How would the solution change if the inequalities had a "less than or equal to" instead of "less than"?
4. Can you solve compound inequalities with absolute values?
5. What happens if both inequalities are "greater than" instead of "less than"?

Tip: When solving compound inequalities, always pay attention to whether the word connecting the inequalities is "and" (intersection of solutions) or "or" (union of solutions).