We are given the compound inequality:

$-8 \leq 5x - 3 < 7$

Let's solve it step by step.

Step 1: Add 3 to all parts of the inequality

To eliminate the constant $-3$ on the middle term, we add 3 to each part of the inequality.

$-8 + 3 \leq 5x - 3 + 3 < 7 + 3$

This simplifies to:

$-5 \leq 5x < 10$

Step 2: Divide by 5

To isolate $x$, we now divide all parts of the inequality by 5.

$\frac{-5}{5} \leq \frac{5x}{5} < \frac{10}{5}$

This simplifies to:

$-1 \leq x < 2$

Final Answer:

The solution is:

$x \in [-1, 2)$

This means $x$ is greater than or equal to $-1$ and strictly less than 2.

Would you like further details on any of the steps? Here are some related questions to deepen your understanding:

1. How do you solve inequalities with variables on both sides?
2. What happens when you divide inequalities by a negative number?
3. Can you graph the solution to this inequality on a number line?
4. How do you solve compound inequalities with absolute values?
5. What is the difference between strict and non-strict inequalities?

Tip: When solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality sign.