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The given inequality is:

\[
-10m + 3 \geq 103 \quad \text{or} \quad 3m - 6 > -27
\]

We need to solve both inequalities.

### Solving the first inequality:

\[
-10m + 3 \geq 103
\]
1. Subtract 3 from both sides:
\[
-10m \geq 100
\]
2. Divide by -10 (remember, when dividing by a negative number, reverse the inequality sign):
\[
m \leq -10
\]

### Solving the second inequality:

\[
3m - 6 > -27
\]
1. Add 6 to both sides:
\[
3m > -21
\]
2. Divide by 3:
\[
m > -7
\]

### Final Answer:

The solution is:
\[
m \leq -10 \quad \text{or} \quad m > -7
\]

This means \( m \) can either be less than or equal to \(-10\), or greater than \(-7\).

Do you need further clarification or have any questions?

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Here are some related questions for practice:
1. How do you solve compound inequalities involving both "and" and "or"?
2. Can you graph the solution to the inequality on a number line?
3. What happens when you multiply or divide both sides of an inequality by a negative number?
4. How do you check if a value satisfies an inequality?
5. How would this problem change if it used "and" instead of "or"?

**Tip**: When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the compound inequality -10m + 3 ≥ 103 or 3m - 6 > -27 with a detailed step-by-step guide. This problem involves solving linear inequalities and understanding the properties of inequalities, making it suitable for students in Grades 8-10.

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