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Let's break down the tax-related questions from the image:

The formula for tax $ T(x) $ is piecewise, depending on the income $ x $:

\[
T(x) = 
\begin{cases} 
0 & \text{if } 0 \leq x \leq 20,000 \\
0.1x - 2,000 & \text{if } 20,000 < x \leq 100,000 \\
0.14x - 6,000 & \text{if } x > 100,000 
\end{cases}
\]

### 1. **Kirk has an income of $10,000. How much will they pay in taxes?**

Kirk's income is $10,000, which falls in the first range: $ 0 \leq x \leq 20,000 $. According to the tax formula, $ T(x) = 0 $ in this range. 

So, Kirk pays **$0** in taxes.

### 2. **Ana has an income of $230,000. How much will they pay in taxes?**

Ana's income is $230,000, which falls in the third range: $ x > 100,000 $. The tax formula for this range is:

\[
T(x) = 0.14x - 6,000
\]

Substituting Ana's income $ x = 230,000 $:

\[
T(230,000) = 0.14(230,000) - 6,000 = 32,200 - 6,000 = 26,200
\]

So, Ana pays **$26,200** in taxes.

### 3. **Dazzline has an income of $90,000. How much will they pay in taxes?**

Dazzline's income is $90,000, which falls in the second range: $ 20,000 < x \leq 100,000 $. The tax formula for this range is:

\[
T(x) = 0.1x - 2,000
\]

Substituting Dazzline's income $ x = 90,000 $:

\[
T(90,000) = 0.1(90,000) - 2,000 = 9,000 - 2,000 = 7,000
\]

So, Dazzline pays **$7,000** in taxes.

### 4. **A politician proposes increasing the tax rate for those making more than $420,000 to 22%. How much more will Kirk pay in taxes?**

Since Kirk's income is $10,000, which is below $420,000, this proposed tax increase would not affect Kirk. Therefore, Kirk will pay **$0 more** in taxes.

---

Would you like more details on any part of the explanation, or have any other questions?

Here are 5 related questions you could consider:

1. What is the total tax paid by someone earning exactly $100,000?
2. How much would Ana pay if the tax rate for incomes above $100,000 were increased to 16% instead of 14%?
3. If Dazzline's income increased to $110,000, how much would they then pay in taxes?
4. How would the tax formula change if the first bracket was taxed at 5% instead of 0%?
5. What would be the tax for someone earning exactly $20,000?

**Tip:** When dealing with piecewise functions, carefully identify which range the input falls into before applying the formula.

In a certain country, the amount of income tax T on earnings of x dollars is computed according to the formula: T(x) = { 0, 0 ≤ x ≤ 20,000; 0.1x - 2000, 20,000 < x ≤ 100,000; 0.14x - 6000, x > 100,000 }. Kirk has an income of $10,000. How much will they pay in taxes? Ana has an income of $230,000. How much will they pay in taxes? Dazzline has an income of $90,000. How much will they pay in taxes? A politician proposes increasing the tax rate, but only on those making more than $420,000, to 22%. How much more will Kirk pay in taxes?

Learn how to calculate income taxes for different income levels using piecewise functions. This problem includes examples such as Kirk with $10,000 income, Ana with $230,000, and Dazzline with $90,000. Discover the tax calculation methods step by step for each scenario.

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