The given compound inequality is:

$-1 < 3x - 4 \leq 14$

Step-by-step solution:

We will solve the compound inequality in two parts. This is essentially solving two inequalities combined into one.

Part 1: Solve $-1 < 3x - 4$

1. Add $4$ to both sides: $-1 + 4 < 3x$ $3 < 3x$

2. Divide both sides by $3$: $1 < x$ $x > 1$

Part 2: Solve $3x - 4 \leq 14$

1. Add $4$ to both sides: $3x - 4 + 4 \leq 14 + 4$ $3x \leq 18$

2. Divide both sides by $3$: $x \leq 6$

Combined Solution:

We now combine the two parts: $1 < x \leq 6$ This means the solution for $x$ is all values between $1$ and $6$, but $x$ is strictly greater than $1$ and less than or equal to $6$.

Graph on the number line:

On the number line:

• There will be an open circle at $x = 1$ (because $x > 1$, not including $1$).
• There will be a closed circle at $x = 6$ (because $x \leq 6$, including $6$).
• A line segment will connect these two points.

Do you have any questions or would you like further details on this solution? Here are 5 related questions to expand your understanding:

1. How would you solve the compound inequality $-3 \leq 2x - 5 < 7$?
2. What does an open circle and closed circle on the number line represent in inequalities?
3. How do you graph the solution of $x > 2$ and $x \leq 8$ on a number line?
4. What is the difference between solving a compound inequality and a simple inequality?
5. Can compound inequalities have more than two parts?

Tip: Always check if the inequality signs are strict (< or >) or inclusive (≤ or ≥) before graphing your solution!