The given inequality is:

\[
-10 > -x - 7 \quad \text{or} \quad -x - 7 > -19
\]

Let's solve each part of the inequality separately.

### Part 1: \(-10 > -x - 7\)

1. Add 7 to both sides:
   \[
   -10 + 7 > -x
   \]
   \[
   -3 > -x
   \]

2. Multiply both sides by \(-1\) (which reverses the inequality sign):
   \[
   3 < x \quad \text{or} \quad x > 3
   \]

### Part 2: \(-x - 7 > -19\)

1. Add 7 to both sides:
   \[
   -x > -19 + 7
   \]
   \[
   -x > -12
   \]

2. Multiply both sides by \(-1\) (which reverses the inequality sign):
   \[
   x < 12
   \]

### Final Solution

The solution to the compound inequality \(x > 3\) or \(x < 12\) depends on how these two inequalities combine. Since it's an "or" condition, we accept all values that satisfy either inequality.

Thus, the solution is:

\[
x > 3 \quad \text{or} \quad x < 12
\]

In this case, since \(x > 3\) or \(x < 12\) covers all values except \(3 \leq x \leq 12\), the final solution can be written as:

\[
(-\infty, 12) \cup (3, \infty)
\]

However, a more straightforward interpretation under "or" is that any value for \(x\) should satisfy at least one of these conditions, which implies \(x\) could be any real number.

### Do you want any more details or have questions?

Here are 5 related questions to further explore this topic:
1. How would the solution change if the inequality was "and" instead of "or"?
2. What are the steps to solve a compound inequality?
3. Can you graph the solution to the inequality on a number line?
4. How does multiplying or dividing by a negative number affect inequalities?
5. What are some real-life situations where inequalities are used?

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

Learn how to solve compound inequalities involving -10 > -x - 7 or -x - 7 > -19 with a detailed step-by-step solution. This problem covers key algebraic concepts, including inequalities and compound inequalities. Suitable for students in Grades 9-12.

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