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We are asked to find the integers that satisfy both of the following inequalities:

1. \( 2x + 5 < 0 \)
2. \( x > -8 \)

### Step 1: Solve the first inequality \( 2x + 5 < 0 \)

Subtract 5 from both sides:

\[
2x < -5
\]

Now divide both sides by 2:

\[
x < -\frac{5}{2} \quad \text{or} \quad x < -2.5
\]

This means \( x \) must be less than -2.5.

### Step 2: Interpret the second inequality \( x > -8 \)

This simply means \( x \) must be greater than -8.

### Step 3: Combine the two inequalities

From the first inequality, we have \( x < -2.5 \), and from the second inequality, we have \( x > -8 \).

Thus, the solution must satisfy:

\[
-8 < x < -2.5
\]

### Step 4: List the integers that satisfy both inequalities

The integers between -8 and -2.5 are:

\[
x = -8, -7, -6, -5, -4, -3
\]

But since \( x \) must be strictly less than -2.5 and greater than -8, the solution is:

\[
x = -7, -6, -5, -4, -3
\]

### Final Answer:
The integers that satisfy both inequalities are \( \{-7, -6, -5, -4, -3\} \).

Would you like more details or have any questions?

Here are 5 follow-up questions related to this:

1. How do we handle strict inequalities like \( < \) or \( > \) in solving inequalities?
2. What is the graphical representation of the solution set for these inequalities?
3. How would the solution change if one of the inequalities was non-strict (e.g., \( \leq \))?
4. Can we solve similar problems using interval notation? How would that look here?
5. How do we check whether an integer satisfies a system of inequalities?

**Tip:** When combining two inequalities, always carefully consider whether they are strict or non-strict inequalities to avoid including or excluding boundary points.

List the integers that satisfy both inequalities: 2x + 5 < 0 and x > -8.

Learn how to solve two linear inequalities and find the integers that satisfy both conditions: 2x + 5 < 0 and x > -8. Detailed step-by-step solution included, ideal for students in Grades 7-9.

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