in a certain country, an individual pays 10% income tax when their monthly income is 1000 or less and when their monthly income is above 1000, they pay 100 plus 15% of the amount in excess of 1000. If M denotes the monthly income of an individual, find all the possible values of M given that the amount of tax they pain was below 1000.

We are given a tax structure where:

- If the monthly income \( M \leq 1000 \), the individual pays 10% tax.
- If the monthly income \( M > 1000 \), the individual pays \( 100 + 15\% \) of the amount in excess of 1000.

We need to find all the possible values of \( M \) such that the amount of tax paid is below 1000.

### Case 1: \( M \leq 1000 \)
In this case, the tax is 10% of \( M \). Therefore, the tax paid \( T \) is given by:
\[
T = 0.1M
\]
We are given that the tax paid is below 1000, so:
\[
0.1M < 1000
\]
Multiplying both sides by 10:
\[
M < 10000
\]
Thus, in this case, \( M \leq 1000 \), which already restricts \( M \), so the solution in this range is \( M \leq 1000 \).

### Case 2: \( M > 1000 \)
In this case, the tax formula is:
\[
T = 100 + 0.15(M - 1000)
\]
We are given that the tax paid is below 1000, so:
\[
100 + 0.15(M - 1000) < 1000
\]
Subtracting 100 from both sides:
\[
0.15(M - 1000) < 900
\]
Now, divide by 0.15:
\[
M - 1000 < 6000
\]
Thus:
\[
M < 7000
\]

### Conclusion
From Case 1, we know that when \( M \leq 1000 \), the condition is satisfied. From Case 2, we found that if \( M > 1000 \), the condition holds when \( M < 7000 \).

Therefore, the possible values of \( M \) are:
\[
M \in (0, 7000)
\]

Would you like more details or have any questions about the solution?

Here are five related questions to expand on this:

1. What would the tax be if the income was exactly 1000?
2. How much tax would be paid if the monthly income is 6000?
3. If the tax paid was exactly 900, what could the income be?
4. How would the tax structure change if the tax rate for income over 1000 increased to 20%?
5. How could we graph the tax paid as a function of income?

**Tip:** Always break down complex tax formulas into simpler conditions based on income ranges for clarity.

In a certain country, an individual pays 10% income tax when their monthly income is 1000 or less and when their monthly income is above 1000, they pay 100 plus 15% of the amount in excess of 1000. If M denotes the monthly income of an individual, find all the possible values of M given that the amount of tax they paid was below 1000.

Explore how to solve a tax calculation problem where an individual's income tax rate changes based on income brackets. The problem involves solving inequalities and piecewise functions to find the range of possible incomes where the tax paid is below 1000. Suitable for students in Grades 10-12.

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