Let's solve the inequality step by step:

We are given the inequality:

$-3 < 2x + 1 \leq 5$

Step 1: Break it into two parts

We can break this compound inequality into two separate inequalities and solve each part:

1. $-3 < 2x + 1$
2. $2x + 1 \leq 5$

Step 2: Solve the first inequality $-3 < 2x + 1$

Subtract 1 from both sides:

$-3 - 1 < 2x$ $-4 < 2x$

Now, divide both sides by 2:

$\frac{-4}{2} < x$ $x > -2$

Step 3: Solve the second inequality $2x + 1 \leq 5$

Subtract 1 from both sides:

$2x \leq 5 - 1$ $2x \leq 4$

Now, divide both sides by 2:

$x \leq \frac{4}{2}$ $x \leq 2$

Step 4: Combine the two results

We now have:

$-2 < x \leq 2$

Step 5: Write the solution in interval notation

The solution set in interval notation is:

$(-2, 2]$

This means that $x$ is strictly greater than $-2$ but less than or equal to $2$.

Do you want more details or have any questions?

Here are 5 related questions:

1. How do you solve inequalities involving absolute values?
2. What is the difference between strict inequalities and non-strict inequalities?
3. Can you explain how to solve inequalities involving quadratic expressions?
4. How does interval notation differ from set notation?
5. What happens when an inequality is multiplied or divided by a negative number?

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