Explanation:
Okay, so I need to solve the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. Hmm, let me think about how to approach this. First, I remember that inequalities involving absolute values can sometimes be tricky because the absolute value makes the expression non-negative. But since there's a fraction here, I also need to consider the denominator and where it's defined.

Alright, let me start by recalling that for a fraction to be less than zero, the numerator and denominator must have opposite signs. Also, the denominator can't be zero because that would make the expression undefined. So, my first step is to figure out where the denominator is zero and where the numerator is zero. Let me write down the denominator and numerator separately.

Denominator: $x^2 - 36x$. Let me factor that. Factoring out an x, I get $x(x - 36)$. So, the denominator is zero when $x = 0$ or $x = 36$. Therefore, these points are excluded from the domain.

Numerator: $|x^2 - 81|$. The absolute value is always non-negative, so the numerator is always greater than or equal to zero. However, since the inequality is strict ($<$), the numerator can't be zero. Therefore, the inequality will never be true when the numerator is zero. Let me check where the numerator is zero. The expression inside the absolute value is $x^2 - 81$, which factors into $(x - 9)(x + 9)$. Therefore, the numerator is zero when $x = 9$ or $x = -9$. So, at these points, the entire fraction is zero, but since the inequality is strict, they are not part of the solution.

So, to recap, the critical points are $x = -9, 0, 9, 36$. These points divide the real number line into intervals where the sign of the expression might change. I need to test each interval to determine where the expression is negative.

But before I proceed, let me note that the numerator is always non-negative, so the sign of the entire fraction depends on the denominator. Since the numerator is non-negative, the fraction will be negative only when the denominator is negative. However, there's a catch here: the numerator is zero at $x = -9$ and $x = 9$, but since the inequality is strict, those points are excluded. Therefore, the solution set will be the intervals where the denominator is negative, excluding points where the numerator is zero. However, I need to check if there are any overlaps.

Wait, actually, the numerator is zero at $x = -9$ and $x = 9$, but the denominator at those points? Let me check:

For $x = -9$: denominator is $(-9)^2 - 36*(-9) = 81 + 324 = 405$, which is positive. So, the fraction is zero here, but since the inequality is strict, we don't include these points.

For $x = 9$: denominator is $9^2 - 36*9 = 81 - 324 = -243$, which is negative. Wait, so at $x = 9$, the numerator is zero (so the entire fraction is zero), but the denominator is negative. Hmm. Wait, if the numerator is zero, the entire fraction is zero regardless of the denominator's sign. So, the fraction is zero at $x = 9$, but since the inequality is strict, we exclude that point. However, even though the denominator is negative at $x = 9$, the fraction is zero there, so that doesn't contribute to the inequality being negative. So, actually, the solution will be the regions where the denominator is negative, but excluding the points where the numerator is zero. Wait, but the numerator is zero at $x = -9$ and $x = 9$. However, at $x = -9$, the denominator is positive, so even though numerator is zero, the fraction is zero. So, the only point where the fraction is zero is $x = -9$ and $x = 9$, but we need to exclude those. However, the denominator is negative in some intervals. Let me try to structure this.

First, find the intervals divided by the critical points $-9$, $0$, $9$, $36$. So, the intervals are:

1. $ (-\infty, -9) $
2. $ (-9, 0) $
3. $ (0, 9) $
4. $ (9, 36) $
5. $ (36, \infty) $

Now, in each interval, I need to determine the sign of the denominator and numerator. However, since the numerator is $|x^2 - 81|$, which is always non-negative, as I said earlier. So, the sign of the fraction depends on the denominator. Remember that the fraction is negative when the denominator is negative (because numerator is non-negative). However, we have to exclude points where the numerator is zero (i.e., $x = -9$ and $x = 9$) because at those points, the fraction is zero, which doesn't satisfy the strict inequality.

But let me check. Wait, if the denominator is negative in some interval, then the fraction is negative there, except where the numerator is zero. But since $|x^2 - 81|$ is zero only at $x = -9$ and $x = 9$, which are in different intervals.

So, let me first find the sign of the denominator $x^2 - 36x$ in each interval.

Denominator: $x^2 - 36x = x(x - 36)$. Let me analyze its sign:

- For $x < 0$: $x$ is negative, $x - 36$ is negative (since x is negative, 36 is positive). So, negative * negative = positive.

- For $0 < x < 36$: $x$ is positive, $x - 36$ is negative. So, positive * negative = negative.

- For $x > 36$: both x and x - 36 are positive, so positive * positive = positive.

Therefore, denominator is positive on $ (-\infty, 0) \cup (36, \infty) $, and negative on $ (0, 36) $. Wait, but the denominator is zero at x=0 and x=36.

Therefore, the denominator is negative only on the interval $ (0, 36) $. However, within that interval, there are points where the numerator is zero, which are at x = -9 and x = 9. Wait, x = -9 is not in the interval (0, 36). x = -9 is in (-infty, -9). Wait, so in the interval (0, 36), the only point where the numerator is zero is x = 9. Therefore, in the interval (0, 36), the numerator is zero at x = 9, and positive elsewhere. Therefore, in the interval (0, 36), the fraction is negative except at x = 9 where it is zero.

Therefore, the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$ is satisfied where denominator is negative, which is (0, 36), except where numerator is zero, which is at x = 9. Therefore, the solution is (0, 9) union (9, 36). But we need to check if there are other intervals where the denominator is negative. Wait, denominator is negative only in (0, 36). So, the answer would be (0, 9) union (9, 36). But let me verify with each interval.

Let me go back to the intervals:

1. $ (-\infty, -9) $: denominator is positive, numerator is |x² - 81|. Since x is less than -9, x² is greater than 81, so x² - 81 is positive, so absolute value is same, so numerator positive, denominator positive: fraction positive. Not less than zero. So, no solution here.

2. $ (-9, 0) $: denominator is positive (as x between -9 and 0, x is negative, x - 36 is negative, so positive). Numerator: |x² - 81|. Since x is between -9 and 0, x² is between 0 and 81. So, x² - 81 is negative, so absolute value makes it positive. Therefore, numerator positive, denominator positive: fraction positive. Not less than zero. No solution here.

3. $ (0, 9) $: denominator is negative (as established earlier). Numerator: |x² - 81|. Here, x between 0 and 9, so x² between 0 and 81, so x² - 81 is negative, absolute value makes it positive. Therefore, numerator positive, denominator negative: fraction negative. So, this interval satisfies the inequality. However, we need to check if there are any points where numerator is zero here. But numerator is zero at x = 9, which is not in this interval. Therefore, entire interval (0, 9) is part of the solution.

4. $ (9, 36) $: denominator is negative. Numerator: |x² - 81|. Here, x between 9 and 36. For x between 9 and 36, x² is greater than 81 (since x > 9), so x² - 81 is positive, so absolute value does not change it. Therefore, numerator is positive, denominator is negative: fraction negative. Therefore, entire interval (9, 36) satisfies the inequality. However, note that at x = 9, numerator is zero. But since x = 9 is excluded (as the inequality is strict), we just exclude that single point.

5. $ (36, \infty) $: denominator positive. Numerator: |x² - 81|. Here, x > 36, so x² is way larger than 81, so x² - 81 positive, absolute value same, so numerator positive, denominator positive: fraction positive. Not less than zero. No solution here.

So, combining the intervals where the inequality holds, we have (0, 9) union (9, 36). But we should also remember that the original expression is undefined at x = 0 and x = 36, so those points are excluded. Also, at x = -9 and x = 9, the expression is zero, which doesn't satisfy the strict inequality, but since x = -9 is not in the intervals where the denominator is negative, it's already excluded because in that interval, the fraction was positive.

Wait, but let me double-check. For x in (0, 9) and (9, 36), the fraction is negative, and there are no other points where it's negative. Therefore, the solution is (0, 9) union (9, 36). However, let me check if there are any other points. For example, is there a possibility that the numerator could be zero in other intervals? Well, the numerator is zero only at x = -9 and x = 9. So, no, those are the only points. Therefore, the solution is indeed (0, 9) union (9, 36).

But let me verify with specific test points.

First, take x = 1, which is in (0,9). Let's compute numerator: |1 - 81| = | -80 | = 80. Denominator: 1 - 36 = -35. So, 80 / (-35) = -80/35 < 0. So, yes, satisfies the inequality.

Take x = 10, which is in (9, 36). Numerator: |100 - 81| = 19. Denominator: 100 - 360 = -260. So, 19 / (-260) < 0. Correct.

Take x = -10, in (-infty, -9). Numerator: |100 - 81| = 19. Denominator: 100 - (-360) = 460. So, 19/460 > 0. Not good.

Take x = -1, in (-9, 0). Numerator: |1 - 81| = 80. Denominator: 1 - (-36) = 37. Positive. So, 80/37 > 0. Not good.

Take x = 37, in (36, ∞). Numerator: |37² - 81| = |1369 - 81| = 1288. Denominator: 37² - 36*37 = 37*(37 - 36) = 37*1 = 37. Positive. So, 1288/37 > 0. Not good.

Take x = 9. At x = 9, numerator is zero, denominator is -243. So, 0 / (-243) = 0, which is not less than zero. So excluded.

Take x = 0: undefined.

x = 36: undefined.

Therefore, the solution set is indeed (0, 9) union (9, 36). So, written in interval notation, that's $ (0, 9) \cup (9, 36) $.

But let me check if there's a possibility that in some interval the numerator might be negative or something? Wait, the numerator is an absolute value, which is always non-negative. Therefore, the only way the fraction is negative is if the denominator is negative. Therefore, as long as denominator is negative, and numerator is not zero (but since numerator is zero at x = -9 and x = 9, but those are already not in the denominator negative interval except x = 9, which is in the denominator negative interval. So at x = 9, the fraction is zero. So we have to exclude that. So the answer is (0,9) union (9,36).

Wait, but perhaps there's another way to approach this problem. Let me try to consider the absolute value expression.

We have |x² - 81|. So, |x² - 81| can be written as:

If x² - 81 ≥ 0, then |x² - 81| = x² - 81.

If x² - 81 < 0, then |x² - 81| = -(x² - 81) = -x² + 81.

Therefore, we can split the inequality into two cases:

Case 1: x² - 81 ≥ 0. That is, |x| ≥ 9, so x ≤ -9 or x ≥ 9.

Case 2: x² - 81 < 0. That is, |x| < 9, so -9 < x < 9.

So, let's analyze each case.

First, the original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$.

Case 1: x ≤ -9 or x ≥ 9. Then |x² - 81| = x² - 81.

So, the inequality becomes $\frac{x^2 - 81}{x^2 - 36x} < 0$.

Note that x cannot be 0 or 36, but in this case, x is ≤ -9 or x ≥ 9, so x = 0 and 36 are not in this case.

So, let me simplify the expression $\frac{x^2 - 81}{x^2 - 36x}$. Factor numerator and denominator:

Numerator: x² - 81 = (x - 9)(x + 9).

Denominator: x² - 36x = x(x - 36).

Therefore, the expression is $\frac{(x - 9)(x + 9)}{x(x - 36)}$.

So, in this case, x ≤ -9 or x ≥ 9. Let's analyze the sign of this expression.

First, note that in Case 1, x ≤ -9 or x ≥ 9.

Let me consider subcases:

Subcase 1a: x ≤ -9.

In this interval, x is negative. Let's consider the factors:

(x - 9): since x ≤ -9, x - 9 is negative (since x ≤ -9 < 9).

(x + 9): since x ≤ -9, x + 9 ≤ 0. So, x + 9 is non-positive. But at x = -9, it is zero. But since x ≤ -9, including x = -9, but x = -9 makes the numerator zero. However, the original inequality is strict, so we have to exclude x = -9. However, in this subcase, we need to check the sign of the expression.

But let me consider x < -9 (excluding x = -9):

(x - 9) is negative.

(x + 9) is negative.

x is negative.

Denominator: x(x - 36). x is negative, x - 36 is negative (since x ≤ -9 < 36). Therefore, denominator is negative * negative = positive.

Therefore, numerator: (negative)*(negative) = positive.

Denominator: positive.

Therefore, overall, expression is positive / positive = positive. Therefore, the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$ is not satisfied here. So, no solution in this subcase.

Subcase 1b: x ≥ 9.

So, x is greater than or equal to 9. Let's consider the factors:

(x - 9): x ≥ 9, so x - 9 ≥ 0.

(x + 9): x ≥ 9, so x + 9 is positive.

x is positive (since x ≥ 9 > 0).

Denominator: x(x - 36). Let's see, x(x - 36):

If x is between 9 and 36, then x - 36 is negative, so denominator is positive * negative = negative.

If x > 36, then x - 36 is positive, so denominator is positive * positive = positive.

So, for x ≥ 9:

Subsubcase 1b1: 9 ≤ x < 36.

Numerator: (x - 9)(x + 9) is positive * positive = positive.

Denominator: negative.

Therefore, expression is positive / negative = negative. Therefore, the inequality holds here. However, note that at x = 9, numerator is zero. Therefore, need to exclude x = 9. Therefore, the interval is [9, 36) but excluding x = 9. So, (9, 36).

Subsubcase 1b2: x > 36.

Numerator: positive (x - 9 and x + 9 positive).

Denominator: positive (x and x - 36 positive). Therefore, expression positive / positive = positive. Therefore, inequality does not hold. So, no solution here.

Therefore, in Case 1, the solution is (9, 36). However, note that at x = 9, the expression is zero, which is excluded. Also, in Subcase 1a, no solution. So, Case 1 contributes (9, 36).

Case 2: -9 < x < 9. In this case, |x² - 81| = -x² + 81. Therefore, the original inequality becomes $\frac{ -x^2 + 81 }{x^2 - 36x} < 0$, which can be written as $\frac{81 - x^2}{x^2 - 36x} < 0$.

Let me simplify this expression. Factor numerator and denominator:

Numerator: 81 - x² = -(x² - 81) = -(x - 9)(x + 9).

Denominator: same as before, x(x - 36).

Therefore, the expression becomes $\frac{ - (x - 9)(x + 9) }{x(x - 36)}$.

Which is equal to $- \frac{(x - 9)(x + 9)}{x(x - 36)}$.

So, we have the inequality $- \frac{(x - 9)(x + 9)}{x(x - 36)} < 0$, which is equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$.

Therefore, in Case 2, we need to find where $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$.

But remember that in Case 2, x is between -9 and 9. So, let's analyze the sign of $\frac{(x - 9)(x + 9)}{x(x - 36)}$ in this interval.

First, note that x is between -9 and 9, so x ≠ 0, and also x ≠ -9 or 9 (which are excluded from the Case 2 interval).

Let me break down Case 2 into subcases.

Subcase 2a: -9 < x < 0.

Subcase 2b: 0 < x < 9.

Let me analyze each.

Subcase 2a: -9 < x < 0.

Factors:

(x - 9): x is negative, so x - 9 is negative.

(x + 9): x is greater than -9, so x + 9 is positive.

x: negative.

(x - 36): x is negative, so x - 36 is negative.

Therefore, numerator: (x - 9)(x + 9) = (negative)(positive) = negative.

Denominator: x(x - 36) = (negative)(negative) = positive.

Therefore, overall fraction is negative / positive = negative. Therefore, $\frac{(x - 9)(x + 9)}{x(x - 36)}$ is negative, so the inequality $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ is false here. Therefore, no solution in Subcase 2a.

Wait, but in Case 2, the original inequality was equivalent to this fraction being positive? Wait, no. Wait, Case 2 led us to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. So, in Subcase 2a, it's negative, so inequality is not satisfied. Therefore, no solution here.

Subcase 2b: 0 < x < 9.

Factors:

(x - 9): negative (since x < 9).

(x + 9): positive (since x > -9, so x + 9 > 0).

x: positive.

(x - 36): negative (since x < 9 < 36).

Therefore, numerator: (x - 9)(x + 9) = (negative)(positive) = negative.

Denominator: x(x - 36) = (positive)(negative) = negative.

Therefore, overall fraction: negative / negative = positive. Therefore, $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ is true here. Therefore, the inequality is satisfied in this interval. However, we need to check if there are any points where the numerator or denominator is zero. In this interval, 0 < x < 9, the numerator (original numerator in Case 2 was |x² - 81| = 81 - x², which is positive here, since x² < 81. Denominator: x² - 36x. For 0 < x < 9, denominator is x(x - 36). Since x is positive and (x - 36) is negative (since x < 36), denominator is negative. Therefore, the original inequality (which was transformed to this fraction being positive) is satisfied here. So, in this interval, the original inequality is satisfied. However, we need to check if the original inequality is satisfied here. Wait, original inequality was $\frac{|x^2 - 81|}{x^2 - 36x} < 0$, which in this case became equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. So, yes, since this fraction is positive, the original inequality is equivalent to that. Therefore, in this interval, the original inequality holds. However, note that in this interval, the numerator |x² - 81| is positive (since x ≠ ±9) and denominator is negative. Therefore, the original fraction is negative, so the inequality holds. So, yes, the interval (0, 9) is part of the solution.

Therefore, in Case 2, the solution is (0, 9).

Therefore, combining Case 1 and Case 2, the solution is (0, 9) union (9, 36).

So, same result as before. Therefore, seems consistent.

Therefore, the answer is (0, 9) union (9, 36). However, to present it properly, need to check if there's an overlap or any mistakes.

Wait, but in Case 1, we had (9, 36) and Case 2, (0, 9). So, combining gives (0, 9) union (9, 36). Therefore, the final answer is all real numbers between 0 and 36, excluding 0, 9, and 36. Wait, but 0 is excluded because denominator is zero there. 9 is excluded because numerator is zero there. 36 is excluded because denominator is zero.

So, yes, the interval is (0, 9) union (9, 36).

But let me check if perhaps there's an error in my Case 1. For example, when I considered Case 1, which is x ≤ -9 or x ≥ 9. Then, in Case 1b, x ≥ 9, we had that the expression is negative in (9, 36). But what about x > 36? There, the expression is positive. So, nothing there. And x ≤ -9 gives positive. So, only (9, 36) from Case 1. And (0, 9) from Case 2. So combined.

Therefore, I think this is correct.

Alternatively, maybe there's another way to think about it. Let me think.

Since the numerator is an absolute value, which is non-negative, the entire fraction is negative only if the denominator is negative and the numerator is not zero. Therefore, the solution is the set of x where denominator is negative, but numerator not zero. Since denominator is negative on (0, 36), and numerator is zero at x = -9 and x = 9. However, x = -9 is not in (0, 36), so only x = 9 is in (0, 36) where numerator is zero. Therefore, the solution is (0, 36) without x = 9, so (0, 9) union (9, 36). Same answer.

Therefore, regardless of the approach, the answer is the same. Therefore, I can be confident.

Hence, the solution is all real numbers between 0 and 36, excluding 0, 9, and 36. Therefore, in interval notation: $ (0, 9) \cup (9, 36) $.

**Final Answer**
The solution to the inequality is \boxed{(0, 9) \cup (9, 36)}.

To solve the inequality
$$
\frac{|x^2 - 81|}{x^2 - 36x} < 0,
$$
we need to carefully analyze both the **numerator** and the **denominator**, keeping in mind the nature of the absolute value and the sign of the expression.

---

### **Step 1: Understand the structure of the expression**

The expression is a **fraction** of the form:

- **Numerator**: $ |x^2 - 81| $, which is **always non-negative** because absolute values are never negative.
- **Denominator**: $ x^2 - 36x = x(x - 36) $, which is defined for all $ x \ne 0 $ and $ x \ne 36 $, and its sign depends on the values of $ x $.

---

### **Step 2: Identify critical points**

We need to find where:

- The **numerator** is zero:
  $$
  |x^2 - 81| = 0 \Rightarrow x^2 - 81 = 0 \Rightarrow x = \pm 9.
  $$
  At these points, the numerator is zero, so the entire expression is zero. Since the inequality is **strict** ($<$), these points are not included in the solution.

- The **denominator** is zero at:
  $$
  x(x - 36) = 0 \Rightarrow x = 0 \text{ or } x = 36.
  $$
  These points are **not included** in the domain (since division by zero is undefined).

So, our **critical points** are:
$$
x = -9,\ 0,\ 9,\ 36.
$$
These divide the real number line into intervals where we will test the sign of the expression.

---

### **Step 3: Analyze the sign of the expression in each interval**

We divide the number line into the following intervals (excluding the critical points) and test the **sign** of the numerator and denominator:

| Interval | Numerator $ |x^2 - 81| $ | Denominator $ x^2 - 36x $ | Sign of Expression |
|----------|------------------|-----------------------------|----------------------|
| $ (-\infty, -9) $ | $> 0$ | $> 0$ | $+ / + = +$ |
| $ (-9, 0) $ | $> 0$ | $> 0$ | $+ / + = +$ |
| $ (0, 9) $ | $> 0$ | $< 0$ | $+ / - = -$ |
| $ (9, 36) $ | $> 0$ | $< 0$ | $+ / - = -$ |
| $ (36, \infty) $ | $> 0$ | $> 0$ | $+ / + = +$ |

We are looking for **where the expression is negative**, so the solution is:
$$
(0, 9) \cup (9, 36)
$$

Note that at $ x = 9 $, the numerator is zero, so the expression is **zero** and does **not** satisfy the strict inequality $ < 0 $. Therefore, we **exclude** $ x = 9 $.

---

### **Step 4: Final Answer**

Combining all valid intervals where the expression is negative and not undefined, the solution is:

$$
\boxed{(0, 9) \cup (9, 36)}
$$

[Assistant]
Explanation:
Okay, so I need to solve the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. Hmm, let me think about how to approach this. First, I remember that inequalities involving absolute values can sometimes be tricky because the absolute value makes the expression non-negative. But since there's a fraction here, I also need to consider the denominator and where it's defined.

Alright, let me start by recalling that for a fraction to be less than zero, the numerator and denominator must have opposite signs. Also, the denominator can't be zero because that would make the expression undefined. So, my first step is to figure out where the denominator is zero and where the numerator is zero. Let me write down the denominator and numerator separately.

Denominator: $x^2 - 36x$. Let me factor that. Factoring out an x, I get $x(x - 36)$. So, the denominator is zero when $x = 0$ or $x = 36$. Therefore, these points are excluded from the domain.

Numerator: $|x^2 - 81|$. The absolute value is always non-negative, so the numerator is always greater than or equal to zero. However, since the inequality is strict ($<$), the numerator can't be zero. Therefore, the inequality will never be true when the numerator is zero. Let me check where the numerator is zero. The expression inside the absolute value is $x^2 - 81$, which factors into $(x - 9)(x + 9)$. Therefore, the numerator is zero when $x = 9$ or $x = -9$. So, at these points, the entire fraction is zero, but since the inequality is strict, they are not part of the solution.

So, to recap, the critical points are $x = -9, 0, 9, 36$. These points divide the real number line into intervals where the sign of the expression might change. I need to test each interval to determine where the expression is negative.

But before I proceed, let me note that the numerator is always non-negative, so the sign of the entire fraction depends on the denominator. Since the numerator is non-negative, the fraction will be negative only when the denominator is negative. However, there's a catch here: the numerator is zero at $x = -9$ and $x = 9$, but since the inequality is strict, those points are excluded. Therefore, the solution set will be the intervals where the denominator is negative, excluding points where the numerator is zero. However, I need to check if there are any overlaps.

Wait, actually, the numerator is zero at $x = -9$ and $x = 9$, but the denominator at those points? Let me check:

For $x = -9$: denominator is $(-9)^2 - 36*(-9) = 81 + 324 = 405$, which is positive. So, the fraction is zero here, but since the inequality is strict, we don't include these points.

For $x = 9$: denominator is $9^2 - 36*9 = 81 - 324 = -243$, which is negative. Wait, so at $x = 9$, the numerator is zero (so the entire fraction is zero), but the denominator is negative. Hmm. Wait, if the numerator is zero, the entire fraction is zero regardless of the denominator's sign. So, the fraction is zero at $x = 9$, but since the inequality is strict, we exclude that point. However, even though the denominator is negative at $x = 9$, the fraction is zero there, so that doesn't contribute to the inequality being negative. So, actually, the solution will be the regions where the denominator is negative, but excluding the points where the numerator is zero. Wait, but the numerator is zero at $x = -9$ and $x = 9$. However, at $x = -9$, the denominator is positive, so even though numerator is zero, the fraction is zero. So, the only point where the fraction is zero is $x = -9$ and $x = 9$, but we need to exclude those. However, the denominator is negative in some intervals. Let me try to structure this.

First, find the intervals divided by the critical points $-9$, $0$, $9$, $36$. So, the intervals are:

1. $ (-\infty, -9) $
2. $ (-9, 0) $
3. $ (0, 9) $
4. $ (9, 36) $
5. $ (36, \infty) $

Now, in each interval, I need to determine the sign of the denominator and numerator. However, since the numerator is $|x^2 - 81|$, which is always non-negative, as I said earlier. So, the sign of the fraction depends on the denominator. Remember that the fraction is negative when the denominator is negative (because numerator is non-negative). However, we have to exclude points where the numerator is zero (i.e., $x = -9$ and $x = 9$) because at those points, the fraction is zero, which doesn't satisfy the strict inequality.

But let me check. Wait, if the denominator is negative in some interval, then the fraction is negative there, except where the numerator is zero. But since $|x^2 - 81|$ is zero only at $x = -9$ and $x = 9$, which are in different intervals.

So, let me first find the sign of the denominator $x^2 - 36x$ in each interval.

Denominator: $x^2 - 36x = x(x - 36)$. Let me analyze its sign:

- For $x < 0$: $x$ is negative, $x - 36$ is negative (since x is negative, 36 is positive). So, negative * negative = positive.

- For $0 < x < 36$: $x$ is positive, $x - 36$ is negative. So, positive * negative = negative.

- For $x > 36$: both x and x - 36 are positive, so positive * positive = positive.

Therefore, denominator is positive on $ (-\infty, 0) \cup (36, \infty) $, and negative on $ (0, 36) $. Wait, but the denominator is zero at x=0 and x=36.

Therefore, the denominator is negative only on the interval $ (0, 36) $. However, within that interval, there are points where the numerator is zero, which are at x = -9 and x = 9. Wait, x = -9 is not in the interval (0, 36). x = -9 is in (-infty, -9). Wait, so in the interval (0, 36), the only point where the numerator is zero is x = 9. Therefore, in the interval (0, 36), the numerator is zero at x = 9, and positive elsewhere. Therefore, in the interval (0, 36), the fraction is negative except at x = 9 where it is zero.

Therefore, the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$ is satisfied where denominator is negative, which is (0, 36), except where numerator is zero, which is at x = 9. Therefore, the solution is (0, 9) union (9, 36). But we need to check if there are other intervals where the denominator is negative. Wait, denominator is negative only in (0, 36). So, the answer would be (0, 9) union (9, 36). But let me verify with each interval.

Let me go back to the intervals:

1. $ (-\infty, -9) $: denominator is positive, numerator is |x² - 81|. Since x is less than -9, x² is greater than 81, so x² - 81 is positive, so absolute value is same, so numerator positive, denominator positive: fraction positive. Not less than zero. So, no solution here.

2. $ (-9, 0) $: denominator is positive (as x between -9 and 0, x is negative, x - 36 is negative, so positive). Numerator: |x² - 81|. Since x is between -9 and 0, x² is between 0 and 81. So, x² - 81 is negative, so absolute value makes it positive. Therefore, numerator positive, denominator positive: fraction positive. Not less than zero. No solution here.

3. $ (0, 9) $: denominator is negative (as established earlier). Numerator: |x² - 81|. Here, x between 0 and 9, so x² between 0 and 81, so x² - 81 is negative, absolute value makes it positive. Therefore, numerator positive, denominator negative: fraction negative. So, this interval satisfies the inequality. However, we need to check if there are any points where numerator is zero here. But numerator is zero at x = 9, which is not in this interval. Therefore, entire interval (0, 9) is part of the solution.

4. $ (9, 36) $: denominator is negative. Numerator: |x² - 81|. Here, x between 9 and 36. For x between 9 and 36, x² is greater than 81 (since x > 9), so x² - 81 is positive, so absolute value does not change it. Therefore, numerator is positive, denominator is negative: fraction negative. Therefore, entire interval (9, 36) satisfies the inequality. However, note that at x = 9, numerator is zero. But since x = 9 is excluded (as the inequality is strict), we just exclude that single point.

5. $ (36, \infty) $: denominator positive. Numerator: |x² - 81|. Here, x > 36, so x² is way larger than 81, so x² - 81 positive, absolute value same, so numerator positive, denominator positive: fraction positive. Not less than zero. No solution here.

So, combining the intervals where the inequality holds, we have (0, 9) union (9, 36). But we should also remember that the original expression is undefined at x = 0 and x = 36, so those points are excluded. Also, at x = -9 and x = 9, the expression is zero, which doesn't satisfy the strict inequality, but since x = -9 is not in the intervals where the denominator is negative, it's already excluded because in that interval, the fraction was positive.

Wait, but let me double-check. For x in (0, 9) and (9, 36), the fraction is negative, and there are no other points where it's negative. Therefore, the solution is (0, 9) union (9, 36). However, let me check if there are any other points. For example, is there a possibility that the numerator could be zero in other intervals? Well, the numerator is zero only at x = -9 and x = 9. So, no, those are the only points. Therefore, the solution is indeed (0, 9) union (9, 36).

But let me verify with specific test points.

First, take x = 1, which is in (0,9). Let's compute numerator: |1 - 81| = | -80 | = 80. Denominator: 1 - 36 = -35. So, 80 / (-35) = -80/35 < 0. So, yes, satisfies the inequality.

Take x = 10, which is in (9, 36). Numerator: |100 - 81| = 19. Denominator: 100 - 360 = -260. So, 19 / (-260) < 0. Correct.

Take x = -10, in (-infty, -9). Numerator: |100 - 81| = 19. Denominator: 100 - (-360) = 460. So, 19/460 > 0. Not good.

Take x = -1, in (-9, 0). Numerator: |1 - 81| = 80. Denominator: 1 - (-36) = 37. Positive. So, 80/37 > 0. Not good.

Take x = 37, in (36, ∞). Numerator: |37² - 81| = |1369 - 81| = 1288. Denominator: 37² - 36*37 = 37*(37 - 36) = 37*1 = 37. Positive. So, 1288/37 > 0. Not good.

Take x = 9. At x = 9, numerator is zero, denominator is -243. So, 0 / (-243) = 0, which is not less than zero. So excluded.

Take x = 0: undefined.

x = 36: undefined.

Therefore, the solution set is indeed (0, 9) union (9, 36). So, written in interval notation, that's $ (0, 9) \cup (9, 36) $.

But let me check if there's a possibility that in some interval the numerator might be negative or something? Wait, the numerator is an absolute value, which is always non-negative. Therefore, the only way the fraction is negative is if the denominator is negative. Therefore, as long as denominator is negative, and numerator is not zero (but since numerator is zero at x = -9 and x = 9, but those are already not in the denominator negative interval except x = 9, which is in the denominator negative interval. So at x = 9, the fraction is zero. So we have to exclude that. So the answer is (0,9) union (9,36).

Wait, but perhaps there's another way to approach this problem. Let me try to consider the absolute value expression.

We have |x² - 81|. So, |x² - 81| can be written as:

If x² - 81 ≥ 0, then |x² - 81| = x² - 81.

If x² - 81 < 0, then |x² - 81| = -(x² - 81) = -x² + 81.

Therefore, we can split the inequality into two cases:

Case 1: x² - 81 ≥ 0. That is, |x| ≥ 9, so x ≤ -9 or x ≥ 9.

Case 2: x² - 81 < 0. That is, |x| < 9, so -9 < x < 9.

So, let's analyze each case.

First, the original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$.

Case 1: x ≤ -9 or x ≥ 9. Then |x² - 81| = x² - 81.

So, the inequality becomes $\frac{x^2 - 81}{x^2 - 36x} < 0$.

Note that x cannot be 0 or 36, but in this case, x is ≤ -9 or x ≥ 9, so x = 0 and 36 are not in this case.

So, let me simplify the expression $\frac{x^2 - 81}{x^2 - 36x}$. Factor numerator and denominator:

Numerator: x² - 81 = (x - 9)(x + 9).

Denominator: x² - 36x = x(x - 36).

Therefore, the expression is $\frac{(x - 9)(x + 9)}{x(x - 36)}$.

So, in this case, x ≤ -9 or x ≥ 9. Let's analyze the sign of this expression.

First, note that in Case 1, x ≤ -9 or x ≥ 9.

Let me consider subcases:

Subcase 1a: x ≤ -9.

In this interval, x is negative. Let's consider the factors:

(x - 9): since x ≤ -9, x - 9 is negative (since x ≤ -9 < 9).

(x + 9): since x ≤ -9, x + 9 ≤ 0. So, x + 9 is non-positive. But at x = -9, it is zero. But since x ≤ -9, including x = -9, but x = -9 makes the numerator zero. However, the original inequality is strict, so we have to exclude x = -9. However, in this subcase, we need to check the sign of the expression.

But let me consider x < -9 (excluding x = -9):

(x - 9) is negative.

(x + 9) is negative.

x is negative.

Denominator: x(x - 36). x is negative, x - 36 is negative (since x ≤ -9 < 36). Therefore, denominator is negative * negative = positive.

Therefore, numerator: (negative)*(negative) = positive.

Denominator: positive.

Therefore, overall, expression is positive / positive = positive. Therefore, the inequality $\frac{|x^2 - 81|}{x^2 - 36x} < 0$ is not satisfied here. So, no solution in this subcase.

Subcase 1b: x ≥ 9.

So, x is greater than or equal to 9. Let's consider the factors:

(x - 9): x ≥ 9, so x - 9 ≥ 0.

(x + 9): x ≥ 9, so x + 9 is positive.

x is positive (since x ≥ 9 > 0).

Denominator: x(x - 36). Let's see, x(x - 36):

If x is between 9 and 36, then x - 36 is negative, so denominator is positive * negative = negative.

If x > 36, then x - 36 is positive, so denominator is positive * positive = positive.

So, for x ≥ 9:

Subsubcase 1b1: 9 ≤ x < 36.

Numerator: (x - 9)(x + 9) is positive * positive = positive.

Denominator: negative.

Therefore, expression is positive / negative = negative. Therefore, the inequality holds here. However, note that at x = 9, numerator is zero. Therefore, need to exclude x = 9. Therefore, the interval is [9, 36) but excluding x = 9. So, (9, 36).

Subsubcase 1b2: x > 36.

Numerator: positive (x - 9 and x + 9 positive).

Denominator: positive (x and x - 36 positive). Therefore, expression positive / positive = positive. Therefore, inequality does not hold. So, no solution here.

Therefore, in Case 1, the solution is (9, 36). However, note that at x = 9, the expression is zero, which is excluded. Also, in Subcase 1a, no solution. So, Case 1 contributes (9, 36).

Case 2: -9 < x < 9. In this case, |x² - 81| = -x² + 81. Therefore, the original inequality becomes $\frac{ -x^2 + 81 }{x^2 - 36x} < 0$, which can be written as $\frac{81 - x^2}{x^2 - 36x} < 0$.

Let me simplify this expression. Factor numerator and denominator:

Numerator: 81 - x² = -(x² - 81) = -(x - 9)(x + 9).

Denominator: same as before, x(x - 36).

Therefore, the expression becomes $\frac{ - (x - 9)(x + 9) }{x(x - 36)}$.

Which is equal to $- \frac{(x - 9)(x + 9)}{x(x - 36)}$.

So, we have the inequality $- \frac{(x - 9)(x + 9)}{x(x - 36)} < 0$, which is equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$.

Therefore, in Case 2, we need to find where $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$.

But remember that in Case 2, x is between -9 and 9. So, let's analyze the sign of $\frac{(x - 9)(x + 9)}{x(x - 36)}$ in this interval.

First, note that x is between -9 and 9, so x ≠ 0, and also x ≠ -9 or 9 (which are excluded from the Case 2 interval).

Let me break down Case 2 into subcases.

Subcase 2a: -9 < x < 0.

Subcase 2b: 0 < x < 9.

Let me analyze each.

Subcase 2a: -9 < x < 0.

Factors:

(x - 9): x is negative, so x - 9 is negative.

(x + 9): x is greater than -9, so x + 9 is positive.

x: negative.

(x - 36): x is negative, so x - 36 is negative.

Therefore, numerator: (x - 9)(x + 9) = (negative)(positive) = negative.

Denominator: x(x - 36) = (negative)(negative) = positive.

Therefore, overall fraction is negative / positive = negative. Therefore, $\frac{(x - 9)(x + 9)}{x(x - 36)}$ is negative, so the inequality $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ is false here. Therefore, no solution in Subcase 2a.

Wait, but in Case 2, the original inequality was equivalent to this fraction being positive? Wait, no. Wait, Case 2 led us to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. So, in Subcase 2a, it's negative, so inequality is not satisfied. Therefore, no solution here.

Subcase 2b: 0 < x < 9.

Factors:

(x - 9): negative (since x < 9).

(x + 9): positive (since x > -9, so x + 9 > 0).

x: positive.

(x - 36): negative (since x < 9 < 36).

Therefore, numerator: (x - 9)(x + 9) = (negative)(positive) = negative.

Denominator: x(x - 36) = (positive)(negative) = negative.

Therefore, overall fraction: negative / negative = positive. Therefore, $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ is true here. Therefore, the inequality is satisfied in this interval. However, we need to check if there are any points where the numerator or denominator is zero. In this interval, 0 < x < 9, the numerator (original numerator in Case 2 was |x² - 81| = 81 - x², which is positive here, since x² < 81. Denominator: x² - 36x. For 0 < x < 9, denominator is x(x - 36). Since x is positive and (x - 36) is negative (since x < 36), denominator is negative. Therefore, the original inequality (which was transformed to this fraction being positive) is satisfied here. So, in this interval, the original inequality is satisfied. However, we need to check if the original inequality is satisfied here. Wait, original inequality was $\frac{|x^2 - 81|}{x^2 - 36x} < 0$, which in this case became equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. So, yes, since this fraction is positive, the original inequality is equivalent to that. Therefore, in this interval, the original inequality holds. However, note that in this interval, the numerator |x² - 81| is positive (since x ≠ ±9) and denominator is negative. Therefore, the original fraction is negative, so the inequality holds. So, yes, the interval (0, 9) is part of the solution.

Therefore, in Case 2, the solution is (0, 9).

Therefore, combining Case 1 and Case 2, the solution is (0, 9) union (9, 36).

So, same result as before. Therefore, seems consistent.

Therefore, the answer is (0, 9) union (9, 36). However, to present it properly, need to check if there's an overlap or any mistakes.

Wait, but in Case 1, we had (9, 36) and Case 2, (0, 9). So, combining gives (0, 9) union (9, 36). Therefore, the final answer is all real numbers between 0 and 36, excluding 0, 9, and 36. Wait, but 0 is excluded because denominator is zero there. 9 is excluded because numerator is zero there. 36 is excluded because denominator is zero.

So, yes, the interval is (0, 9) union (9, 36).

But let me check if perhaps there's an error in my Case 1. For example, when I considered Case 1, which is x ≤ -9 or x ≥ 9. Then, in Case 1b, x ≥ 9, we had that the expression is negative in (9, 36). But what about x > 36? There, the expression is positive. So, nothing there. And x ≤ -9 gives positive. So, only (9, 36) from Case 1. And (0, 9) from Case 2. So combined.

Therefore, I think this is correct.

Alternatively, maybe there's another way to think about it. Let me think.

Since the numerator is an absolute value, which is non-negative, the entire fraction is negative only if the denominator is negative and the numerator is not zero. Therefore, the solution is the set of x where denominator is negative, but numerator not zero. Since denominator is negative on (0, 36), and numerator is zero at x = -9 and x = 9. However, x = -9 is not in (0, 36), so only x = 9 is in (0, 36) where numerator is zero. Therefore, the solution is (0, 36) without x = 9, so (0, 9) union (9, 36). Same answer.

Therefore, regardless of the approach, the answer is the same. Therefore, I can be confident.

Hence, the solution is all real numbers between 0 and 36, excluding 0, 9, and 36. Therefore, in interval notation: $ (0, 9) \cup (9, 36) $.

**Final Answer**
The solution to the inequality is \boxed{(0, 9) \cup (9, 36)}.

To solve the inequality
$$
\frac{|x^2 - 81|}{x^2 - 36x} < 0,
$$
we need to carefully analyze both the **numerator** and the **denominator**, keeping in mind the nature of the absolute value and the sign of the expression.

---

### **Step 1: Understand the structure of the expression**

The expression is a **fraction** of the form:

- **Numerator**: $ |x^2 - 81| $, which is **always non-negative** because absolute values are never negative.
- **Denominator**: $ x^2 - 36x = x(x - 36) $, which is defined for all $ x \ne 0 $ and $ x \ne 36 $, and its sign depends on the values of $ x $.

---

### **Step 2: Identify critical points**

We need to find where:

- The **numerator** is zero:
  $$
  |x^2 - 81| = 0 \Rightarrow x^2 - 81 = 0 \Rightarrow x = \pm 9.
  $$
  At these points, the numerator is zero, so the entire expression is zero. Since the inequality is **strict** ($<$), these points are not included in the solution.

- The **denominator** is zero at:
  $$
  x(x - 36) = 0 \Rightarrow x = 0 \text{ or } x = 36.
  $$
  These points are **not included** in the domain (since division by zero is undefined).

So, our **critical points** are:
$$
x = -9,\ 0,\ 9,\ 36.
$$
These divide the real number line into intervals where we will test the sign of the expression.

---

### **Step 3: Analyze the sign of the expression in each interval**

We divide the number line into the following intervals (excluding the critical points) and test the **sign** of the numerator and denominator:

| Interval | Numerator $ |x^2 - 81| $ | Denominator $ x^2 - 36x $ | Sign of Expression |
|----------|------------------|-----------------------------|----------------------|
| $ (-\infty, -9) $ | $> 0$ | $> 0$ | $+ / + = +$ |
| $ (-9, 0) $ | $> 0$ | $> 0$ | $+ / + = +$ |
| $ (0, 9) $ | $> 0$ | $< 0$ | $+ / - = -$ |
| $ (9, 36) $ | $> 0$ | $< 0$ | $+ / - = -$ |
| $ (36, \infty) $ | $> 0$ | $> 0$ | $+ / + = +$ |

We are looking for **where the expression is negative**, so the solution is:
$$
(0, 9) \cup (9, 36)
$$

Note that at $ x = 9 $, the numerator is zero, so the expression is **zero** and does **not** satisfy the strict inequality $ < 0 $. Therefore, we **exclude** $ x = 9 $.

---

### **Step 4: Final Answer**

Combining all valid intervals where the expression is negative and not undefined, the solution is:

$$
\boxed{(0, 9) \cup (9, 36)}
$$

Solution:
Note that $|x^2 - 81| \ge 0$ for all $x,$ with $|x^2 - 81| = 0$ only for $x = \pm 9.$

The denominator factors as $x(x - 36).$  This is negative only for $0 < x < 36.$  Thus, the solution is
\[x \in \boxed{(0,9) \cup (9,36)}.\]


Explanation:
Okay, so I need to solve the inequality $ x^2 - 5x - 4 \leq 10 $. Let me think about how to approach this.

First, I remember that solving quadratic inequalities usually involves moving all terms to one side so that the inequality is set to zero. That way, I can analyze the quadratic function and determine where it satisfies the inequality. Let me try that here.

Starting with the original inequality:
$$ x^2 - 5x - 4 \leq 10 $$

I need to subtract 10 from both sides to get everything on the left:
$$ x^2 - 5x - 4 - 10 \leq 0 $$
Simplifying the constants:
$$ x^2 - 5x - 14 \leq 0 $$

Okay, so now the inequality is $ x^2 - 5x - 14 \leq 0 $. Next, I should find the roots of the quadratic equation $ x^2 - 5x - 14 = 0 $ because the solutions to the inequality will depend on where the quadratic is below or equal to zero.

To find the roots, I can use the quadratic formula. The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = -5 $, and $ c = -14 $. Let me plug those values in.

Calculating the discriminant first:
$$ b^2 - 4ac = (-5)^2 - 4(1)(-14) $$
$$ = 25 - 4(1)(-14) $$
$$ = 25 + 56 $$
$$ = 81 $$

Oh, the discriminant is 81, which is a perfect square, so the roots will be rational numbers. That's good because it makes things easier.

Now applying the quadratic formula:
$$ x = \frac{-(-5) \pm \sqrt{81}}{2(1)} $$
$$ x = \frac{5 \pm 9}{2} $$

So, there are two solutions here:
1. $ x = \frac{5 + 9}{2} = \frac{14}{2} = 7 $
2. $ x = \frac{5 - 9}{2} = \frac{-4}{2} = -2 $

Therefore, the roots are $ x = -2 $ and $ x = 7 $.

Now, since the quadratic coefficient $ a = 1 $ is positive, the parabola opens upwards. That means the quadratic will be below or equal to zero between its two roots. Wait, let me make sure I remember this correctly. If a quadratic opens upwards (since $ a > 0 $), it is negative between its two roots and positive outside. So, the inequality $ x^2 - 5x - 14 \leq 0 $ would hold for $ x $ between -2 and 7, including the endpoints because of the "less than or equal to" sign.

Therefore, the solution in interval notation should be $ [-2, 7] $.

But let me double-check to ensure I didn't make any mistakes. Let me test a value in each interval to confirm.

First, let me check a value less than -2, say x = -3:
Original inequality: $ (-3)^2 -5(-3) -4 \leq 10 $
Calculating:
$ 9 + 15 - 4 = 20 $. Is 20 ≤ 10? No. So that's outside the solution set, which matches our interval.

Next, check a value between -2 and 7, say x = 0:
Original inequality: $ 0^2 -5(0) -4 \leq 10 $
Which is -4 ≤ 10. Yes, that's true.

Now check a value greater than 7, say x = 8:
Original inequality: $ 8^2 -5(8) -4 \leq 10 $
Calculating:
64 - 40 -4 = 20. Is 20 ≤ 10? No. So that's outside, which matches our interval.

What about the endpoints? Let's check x = -2:
Left side: (-2)^2 -5(-2) -4 = 4 +10 -4 = 10. 10 ≤ 10 is true.

x = 7:
7^2 -5(7) -4 = 49 - 35 -4 = 10. Again, 10 ≤ 10 is true. So endpoints are included.

Therefore, the solution set is indeed from -2 to 7, including both ends. So interval notation is [-2, 7].

Wait, let me just make sure that when I moved the 10 to the left, I did it correctly. Original inequality was $ x^2 -5x -4 \leq 10 $, then subtract 10: $ x^2 -5x -4 -10 \leq 0 $, which is $ x^2 -5x -14 \leq 0 $. Yep, that seems right.

Alternatively, maybe I can think about the original inequality as $ x^2 -5x -4 -10 \leq 0 $, which simplifies to the same thing.

Another thing I can check is whether the quadratic was factored correctly. Wait, the quadratic equation was $ x^2 -5x -14 $. Let me see if it factors.

Looking for two numbers that multiply to -14 and add to -5. Let me think. Factors of -14 could be 2 and -7. 2 * (-7) = -14. 2 + (-7) = -5. Yes! So the quadratic factors as (x + 2)(x - 7). Let me check that:

Multiply (x + 2)(x - 7) = x^2 -7x + 2x -14 = x^2 -5x -14. Perfect, that's correct.

So, the inequality is $ (x + 2)(x - 7) \leq 0 $.

This is helpful because when we have a product of two terms less than or equal to zero, the product is negative or zero when one term is positive and the other is negative, or when one or both are zero.

So, the critical points are at x = -2 and x = 7.

To determine where the product is negative or zero, we can use a sign chart.

For x < -2, let's pick x = -3. Then (x + 2) = -1 (negative), (x - 7) = -10 (negative). Negative times negative is positive. So positive, which doesn't satisfy the inequality.

Between -2 and 7, let's pick x = 0. (x + 2) = 2 (positive), (x - 7) = -7 (negative). Positive times negative is negative, which satisfies the inequality.

For x > 7, say x = 8. (x + 2) = 10 (positive), (x - 7) = 1 (positive). Positive times positive is positive, which doesn't satisfy the inequality.

At the critical points: x = -2, one term is zero, so the product is zero. Similarly for x = 7. So those are included in the solution.

Therefore, the solution is indeed all real numbers from -2 to 7, including endpoints. So interval notation is [-2, 7].

I think that's solid. I've checked multiple ways: solving via quadratic formula, factoring, testing points, and using sign analysis. All methods lead to the same conclusion.

Therefore, the answer should be \boxed{[-2, 7]}.

**Final Answer**
The solution to the inequality is \boxed{[-2, 7]}.

To solve the inequality
$$
x^2 - 5x - 4 \leq 10,
$$
we begin by moving all terms to one side to express the inequality in standard quadratic form:

$$
x^2 - 5x - 4 - 10 \leq 0
$$
$$
x^2 - 5x - 14 \leq 0.
$$

---

### Step 1: Solve the Corresponding Quadratic Equation

We solve the equation
$$
x^2 - 5x - 14 = 0
$$
using the quadratic formula:

$$
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}
= \frac{5 \pm \sqrt{25 + 56}}{2}
= \frac{5 \pm \sqrt{81}}{2}
= \frac{5 \pm 9}{2}.
$$

This gives the two roots:

$$
x = \frac{5 + 9}{2} = \frac{14}{2} = 7, \quad x = \frac{5 - 9}{2} = \frac{-4}{2} = -2.
$$

So, the roots are $ x = -2 $ and $ x = 7 $.

---

### Step 2: Analyze the Sign of the Quadratic Expression

We write the inequality as:

$$
x^2 - 5x - 14 \leq 0,
$$
which can be factored as:

$$
(x + 2)(x - 7) \leq 0.
$$

We now analyze the sign of the product $ (x + 2)(x - 7) $ by considering the intervals determined by the roots:

- $ x < -2 $
- $ -2 < x < 7 $
- $ x > 7 $

#### Interval 1: $ x < -2 $
Pick $ x = -3 $:
$ (x+2) = -1 $, $ (x-7) = -10 $, so product = $ (-1)(-10) = 10 > 0 $
→ **Does not satisfy** the inequality

#### Interval 2: $ -2 < x < 7 $
Pick $ x = 0 $:
$ (x+2) = 2 $, $ (x-7) = -7 $, so product = $ (2)(-7) = -14 < 0 $
→ **Satisfies** the inequality

#### Interval 3: $ x > 7 $
Pick $ x = 8 $:
$ (x+2) = 10 $, $ (x-7) = 1 $, so product = $ (10)(1) = 10 > 0 $
→ **Does not satisfy** the inequality

At the **critical points**:
- At $ x = -2 $, one factor is zero: $ (x + 2) = 0 $, so product = 0 → **included**
- At $ x = 7 $, one factor is zero: $ (x - 7) = 0 $, so product = 0 → **included**

---

### Final Solution

Thus, the inequality $ (x + 2)(x - 7) \leq 0 $ is satisfied for all values of $ x $ in the interval:

$$
[-2, 7]
$$

---

### Final Answer

$$
\boxed{[-2, 7]}
$$

[Assistant]
Explanation:
Okay, so I need to solve the inequality $ x^2 - 5x - 4 \leq 10 $. Let me think about how to approach this.

First, I remember that solving quadratic inequalities usually involves moving all terms to one side so that the inequality is set to zero. That way, I can analyze the quadratic function and determine where it satisfies the inequality. Let me try that here.

Starting with the original inequality:
$$ x^2 - 5x - 4 \leq 10 $$

I need to subtract 10 from both sides to get everything on the left:
$$ x^2 - 5x - 4 - 10 \leq 0 $$
Simplifying the constants:
$$ x^2 - 5x - 14 \leq 0 $$

Okay, so now the inequality is $ x^2 - 5x - 14 \leq 0 $. Next, I should find the roots of the quadratic equation $ x^2 - 5x - 14 = 0 $ because the solutions to the inequality will depend on where the quadratic is below or equal to zero.

To find the roots, I can use the quadratic formula. The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = -5 $, and $ c = -14 $. Let me plug those values in.

Calculating the discriminant first:
$$ b^2 - 4ac = (-5)^2 - 4(1)(-14) $$
$$ = 25 - 4(1)(-14) $$
$$ = 25 + 56 $$
$$ = 81 $$

Oh, the discriminant is 81, which is a perfect square, so the roots will be rational numbers. That's good because it makes things easier.

Now applying the quadratic formula:
$$ x = \frac{-(-5) \pm \sqrt{81}}{2(1)} $$
$$ x = \frac{5 \pm 9}{2} $$

So, there are two solutions here:
1. $ x = \frac{5 + 9}{2} = \frac{14}{2} = 7 $
2. $ x = \frac{5 - 9}{2} = \frac{-4}{2} = -2 $

Therefore, the roots are $ x = -2 $ and $ x = 7 $.

Now, since the quadratic coefficient $ a = 1 $ is positive, the parabola opens upwards. That means the quadratic will be below or equal to zero between its two roots. Wait, let me make sure I remember this correctly. If a quadratic opens upwards (since $ a > 0 $), it is negative between its two roots and positive outside. So, the inequality $ x^2 - 5x - 14 \leq 0 $ would hold for $ x $ between -2 and 7, including the endpoints because of the "less than or equal to" sign.

Therefore, the solution in interval notation should be $ [-2, 7] $.

But let me double-check to ensure I didn't make any mistakes. Let me test a value in each interval to confirm.

First, let me check a value less than -2, say x = -3:
Original inequality: $ (-3)^2 -5(-3) -4 \leq 10 $
Calculating:
$ 9 + 15 - 4 = 20 $. Is 20 ≤ 10? No. So that's outside the solution set, which matches our interval.

Next, check a value between -2 and 7, say x = 0:
Original inequality: $ 0^2 -5(0) -4 \leq 10 $
Which is -4 ≤ 10. Yes, that's true.

Now check a value greater than 7, say x = 8:
Original inequality: $ 8^2 -5(8) -4 \leq 10 $
Calculating:
64 - 40 -4 = 20. Is 20 ≤ 10? No. So that's outside, which matches our interval.

What about the endpoints? Let's check x = -2:
Left side: (-2)^2 -5(-2) -4 = 4 +10 -4 = 10. 10 ≤ 10 is true.

x = 7:
7^2 -5(7) -4 = 49 - 35 -4 = 10. Again, 10 ≤ 10 is true. So endpoints are included.

Therefore, the solution set is indeed from -2 to 7, including both ends. So interval notation is [-2, 7].

Wait, let me just make sure that when I moved the 10 to the left, I did it correctly. Original inequality was $ x^2 -5x -4 \leq 10 $, then subtract 10: $ x^2 -5x -4 -10 \leq 0 $, which is $ x^2 -5x -14 \leq 0 $. Yep, that seems right.

Alternatively, maybe I can think about the original inequality as $ x^2 -5x -4 -10 \leq 0 $, which simplifies to the same thing.

Another thing I can check is whether the quadratic was factored correctly. Wait, the quadratic equation was $ x^2 -5x -14 $. Let me see if it factors.

Looking for two numbers that multiply to -14 and add to -5. Let me think. Factors of -14 could be 2 and -7. 2 * (-7) = -14. 2 + (-7) = -5. Yes! So the quadratic factors as (x + 2)(x - 7). Let me check that:

Multiply (x + 2)(x - 7) = x^2 -7x + 2x -14 = x^2 -5x -14. Perfect, that's correct.

So, the inequality is $ (x + 2)(x - 7) \leq 0 $.

This is helpful because when we have a product of two terms less than or equal to zero, the product is negative or zero when one term is positive and the other is negative, or when one or both are zero.

So, the critical points are at x = -2 and x = 7.

To determine where the product is negative or zero, we can use a sign chart.

For x < -2, let's pick x = -3. Then (x + 2) = -1 (negative), (x - 7) = -10 (negative). Negative times negative is positive. So positive, which doesn't satisfy the inequality.

Between -2 and 7, let's pick x = 0. (x + 2) = 2 (positive), (x - 7) = -7 (negative). Positive times negative is negative, which satisfies the inequality.

For x > 7, say x = 8. (x + 2) = 10 (positive), (x - 7) = 1 (positive). Positive times positive is positive, which doesn't satisfy the inequality.

At the critical points: x = -2, one term is zero, so the product is zero. Similarly for x = 7. So those are included in the solution.

Therefore, the solution is indeed all real numbers from -2 to 7, including endpoints. So interval notation is [-2, 7].

I think that's solid. I've checked multiple ways: solving via quadratic formula, factoring, testing points, and using sign analysis. All methods lead to the same conclusion.

Therefore, the answer should be \boxed{[-2, 7]}.

**Final Answer**
The solution to the inequality is \boxed{[-2, 7]}.

To solve the inequality
$$
x^2 - 5x - 4 \leq 10,
$$
we begin by moving all terms to one side to express the inequality in standard quadratic form:

$$
x^2 - 5x - 4 - 10 \leq 0
$$
$$
x^2 - 5x - 14 \leq 0.
$$

---

### Step 1: Solve the Corresponding Quadratic Equation

We solve the equation
$$
x^2 - 5x - 14 = 0
$$
using the quadratic formula:

$$
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}
= \frac{5 \pm \sqrt{25 + 56}}{2}
= \frac{5 \pm \sqrt{81}}{2}
= \frac{5 \pm 9}{2}.
$$

This gives the two roots:

$$
x = \frac{5 + 9}{2} = \frac{14}{2} = 7, \quad x = \frac{5 - 9}{2} = \frac{-4}{2} = -2.
$$

So, the roots are $ x = -2 $ and $ x = 7 $.

---

### Step 2: Analyze the Sign of the Quadratic Expression

We write the inequality as:

$$
x^2 - 5x - 14 \leq 0,
$$
which can be factored as:

$$
(x + 2)(x - 7) \leq 0.
$$

We now analyze the sign of the product $ (x + 2)(x - 7) $ by considering the intervals determined by the roots:

- $ x < -2 $
- $ -2 < x < 7 $
- $ x > 7 $

#### Interval 1: $ x < -2 $
Pick $ x = -3 $:
$ (x+2) = -1 $, $ (x-7) = -10 $, so product = $ (-1)(-10) = 10 > 0 $
→ **Does not satisfy** the inequality

#### Interval 2: $ -2 < x < 7 $
Pick $ x = 0 $:
$ (x+2) = 2 $, $ (x-7) = -7 $, so product = $ (2)(-7) = -14 < 0 $
→ **Satisfies** the inequality

#### Interval 3: $ x > 7 $
Pick $ x = 8 $:
$ (x+2) = 10 $, $ (x-7) = 1 $, so product = $ (10)(1) = 10 > 0 $
→ **Does not satisfy** the inequality

At the **critical points**:
- At $ x = -2 $, one factor is zero: $ (x + 2) = 0 $, so product = 0 → **included**
- At $ x = 7 $, one factor is zero: $ (x - 7) = 0 $, so product = 0 → **included**

---

### Final Solution

Thus, the inequality $ (x + 2)(x - 7) \leq 0 $ is satisfied for all values of $ x $ in the interval:

$$
[-2, 7]
$$

---

### Final Answer

$$
\boxed{[-2, 7]}
$$

Solution: Re-arranging, $x^2 - 5x - 14 \le 0$. The left-hand quadratic factors as $x^2 - 5x - 14 = (x - 7)(x + 2) \le 0$. Thus, $x-7$ and $x+2$ have opposite signs, so $-2 \le x \le 7$ and $\boxed{x \in [-2,7]}$.


Explanation:
Okay, let's see. Navin, Luka, and Ian are from different countries, and they each have different hourly wages in their local currencies. The question is asking who would earn the most after working an eight-hour day if we convert their wages to US dollars.

First, let me list out the given information to make sure I have everything straight.

Navin from Mauritius makes 160 rupee per hour. Luka from Croatia makes 25 kuna per hour. Ian from Botswana makes 34 pula per hour.

The exchange rates are:
1 US dollar = 32.35 Mauritius rupee
1 US dollar = 5.18 Croatian kuna
1 US dollar = 6.95 Botswana pula

So, the plan is to convert each of their hourly wages into US dollars, then multiply by 8 hours to get the total earnings for the day, and then compare those totals. The person with the highest total in USD would be the one earning the most.

Let me start with Navin. He makes 160 rupee per hour. Since 1 USD is 32.35 rupee, I need to find out how many USD 160 rupee is. To do that, I can divide his hourly wage by the exchange rate.

So Navin's hourly wage in USD is 160 / 32.35. Let me calculate that.

First, let me write down the calculation: 160 divided by 32.35. Let me get a calculator for this... Wait, maybe I can approximate it? Or is there a better way?

Alternatively, maybe I can compute it step by step. Let me do this division. Let me think. 32.35 times 4 is 129.4. Then 32.35 times 5 is 161.75. Oh, so 32.35 times 5 is 161.75, which is just a bit more than 160. So Navin's wage is slightly less than 5 USD per hour. Let me compute the exact value.

So 160 divided by 32.35. Let me write it as 160 / 32.35. Let me convert this to decimal. Let me multiply numerator and denominator by 100 to eliminate decimals: 16000 / 3235. Let me approximate this division.

Alternatively, since I know that 32.35 * 4.95 = let's see... Well, maybe it's easier to just do decimal division.

Let me try:

32.35 goes into 160 how many times?

Well, 32.35 * 4 = 129.4

Subtract that from 160: 160 - 129.4 = 30.6

Bring down a zero (assuming decimal division). Wait, since it's 160 divided by 32.35, maybe I should write it as 160.00 divided by 32.35.

So 32.35 goes into 160.00 four times (as before), with remainder 30.6. Now, add a decimal point and continue.

So 32.35 goes into 306.0 (after multiplying remainder by 10) how many times? Let's see, 32.35 * 9 = 291.15, which is less than 306. So 9 times. Subtract 291.15 from 306.0: 306.0 - 291.15 = 14.85. Bring down a zero, making it 148.5.

Now, 32.35 goes into 148.5 about 4 times (since 32.35 * 4 = 129.4). Subtract that from 148.5: 148.5 - 129.4 = 19.1. Bring down another zero, making it 191.0.

32.35 goes into 191.0 about 5 times (32.35 * 5 = 161.75). Subtract that: 191.0 - 161.75 = 29.25. Bring down another zero, making it 292.5.

32.35 goes into 292.5 about 9 times (since 32.35 * 9 = 291.15). Subtract that: 292.5 - 291.15 = 1.35.

So putting it all together, we had 4.9 (from the first two steps: 4 and then 9), then 4, then 5, then 9... So approximately 4.9459... So roughly $4.95 per hour.

Wait, let me check that again. Wait, initial division:

First step: 4. Then decimal, then 9, 4, 5, 9... So 4.945... So approximately $4.95 per hour.

But let me check with calculator steps. Let me use another approach. Let me compute 160 / 32.35.

Let me approximate 32.35 as 32.35. Let me note that 32.35 * 5 = 161.75, which is 1.75 more than 160. So, 5 - (1.75 / 32.35). So approximately 5 - 0.054 = 4.946. So yeah, about $4.946 per hour.

So Navin's hourly wage in USD is approximately $4.95.

Now, moving on to Luka. He makes 25 kuna per hour. The exchange rate is 1 USD = 5.18 kuna. Therefore, to find his hourly wage in USD, we divide his wage by 5.18.

So 25 / 5.18. Let me compute this. Let me think.

Again, 5.18 times 4 is 20.72. Subtract that from 25: 25 - 20.72 = 4.28. Then, 5.18 goes into 4.28 about 0.825 times? Let me see.

Alternatively, let me do decimal division. 25 divided by 5.18. Multiply numerator and denominator by 100 to get rid of decimals: 2500 / 518. Let me approximate this.

518 * 4 = 2072. 2500 - 2072 = 428. Bring down a zero (making it 4280). 518 * 8 = 4144. Subtract from 4280: 4280 - 4144 = 136. Bring down another zero, making it 1360. 518 * 2 = 1036. Subtract: 1360 - 1036 = 324. Bring down another zero, making it 3240. 518 * 6 = 3108. Subtract: 3240 - 3108 = 132.

So so far, we have 4.826... So approximately $4.83 per hour. Let me check with calculator steps. Since 5.18 * 4.8 = let's compute that. 5 * 4.8 = 24, 0.18 * 4.8 = 0.864. Total is 24.864. Which is less than 25. The difference is 25 - 24.864 = 0.136. So then 0.136 / 5.18 ≈ 0.0263. So total is approximately 4.8 + 0.0263 ≈ 4.8263. So yeah, approximately $4.83 per hour. So Luka's hourly wage is roughly $4.83.

Now, Ian from Botswana makes 34 pula per hour. The exchange rate is 1 USD = 6.95 pula. Therefore, his hourly wage in USD is 34 divided by 6.95. Let me compute that.

Again, let me do decimal division. 34 / 6.95. Multiply numerator and denominator by 100: 3400 / 695. Let me simplify that.

First, note that 695 * 4 = 2780. Subtract that from 3400: 3400 - 2780 = 620. Bring down a zero (making it 6200). 695 * 8 = 5560. Subtract: 6200 - 5560 = 640. Bring down another zero, making it 6400. 695 * 9 = 6255. Subtract: 6400 - 6255 = 145. Bring down another zero, making it 1450. 695 * 2 = 1390. Subtract: 1450 - 1390 = 60.

So so far, we have 4.892... So approximately $4.89 per hour. Let me check with another method. Let's see, 6.95 * 4.8 = ?

First, 6.95 * 4 = 27.8, 6.95 * 0.8 = 5.56. Total is 27.8 + 5.56 = 33.36. Which is less than 34. The difference is 34 - 33.36 = 0.64. Then 0.64 / 6.95 ≈ 0.092. So total is 4.8 + 0.092 ≈ 4.892. So yes, approximately $4.89 per hour. So Ian's hourly wage is roughly $4.89 per hour.

So summarizing:

Navin: ~$4.95 per hour

Luka: ~$4.83 per hour

Ian: ~$4.89 per hour

So Navin has the highest hourly wage in USD. Therefore, after working 8 hours, Navin would earn the most. But let me check again to make sure I didn't make a mistake in calculations.

Wait, but let me verify the exact numbers again. Maybe my approximations are off? Let me compute each one more precisely.

Starting with Navin: 160 / 32.35. Let me use a calculator-like approach. Let me note that 32.35 * 4.9 = ?

Well, 32.35 * 4 = 129.4

32.35 * 0.9 = 29.115

Adding those gives 129.4 + 29.115 = 158.515. Which is less than 160. The difference is 160 - 158.515 = 1.485. So 1.485 / 32.35 ≈ 0.0459. Therefore, total is 4.9 + 0.0459 ≈ 4.9459. So $4.9459 per hour.

So Navin's hourly rate is approximately $4.9459.

For Luka: 25 / 5.18. Let me compute that. Let me consider that 5.18 * 4.8 = 24.864 (as before). The difference between 25 and 24.864 is 0.136. Then 0.136 / 5.18 ≈ 0.0263. So total is 4.8 + 0.0263 ≈ 4.8263. So approximately $4.8263 per hour.

For Ian: 34 / 6.95. Let me compute 6.95 * 4.89 = ?

Well, 6.95 * 4 = 27.8

6.95 * 0.89 = ?

First, 6.95 * 0.8 = 5.56

6.95 * 0.09 = 0.6255

Adding those gives 5.56 + 0.6255 = 6.1855

So total is 27.8 + 6.1855 = 33.9855, which is very close to 34. The difference is 34 - 33.9855 = 0.0145. Therefore, 0.0145 / 6.95 ≈ 0.002086. Therefore, total rate is approximately 4.89 + 0.002086 ≈ 4.8921. So approximately $4.8921 per hour.

Therefore, precise rates:

Navin: ~$4.9459

Luka: ~$4.8263

Ian: ~$4.8921

So Navin still has the highest hourly rate. Therefore, over 8 hours, their earnings would be:

Navin: 4.9459 * 8

Luka: 4.8263 * 8

Ian: 4.8921 * 8

Let me compute each of those.

Starting with Navin:

4.9459 * 8. Let me compute 4.9459 * 8.

Well, 4 * 8 = 32

0.9459 * 8 = ?

0.9 * 8 = 7.2

0.0459 * 8 = 0.3672

So total is 7.2 + 0.3672 = 7.5672

Therefore, total earnings: 32 + 7.5672 = 39.5672 USD. Approximately $39.57.

Luka: 4.8263 * 8.

Let me compute that. 4 * 8 = 32. 0.8263 * 8 = ?

0.8 * 8 = 6.4

0.0263 * 8 = 0.2104

Total: 6.4 + 0.2104 = 6.6104

So total earnings: 32 + 6.6104 = 38.6104 USD. Approximately $38.61.

Ian: 4.8921 * 8.

Compute 4 * 8 = 32. 0.8921 * 8.

0.8 * 8 = 6.4

0.0921 * 8 = 0.7368

Total: 6.4 + 0.7368 = 7.1368

Therefore, total earnings: 32 + 7.1368 = 39.1368 USD. Approximately $39.14.

So Navin earns approximately $39.57, Luka $38.61, Ian $39.14. Therefore, Navin earns the most.

But wait, let me check my calculations again to make sure.

For Navin: 160 rupee per hour. 1 USD = 32.35 rupee. So 160 / 32.35. Let me compute 160 / 32.35 with more precision. Let me do 160 / 32.35.

Let me note that 32.35 * 4.94 = ?

Let me compute 32.35 * 4 = 129.4

32.35 * 0.94 = ?

First, 32.35 * 0.9 = 29.115

32.35 * 0.04 = 1.294

Adding them: 29.115 + 1.294 = 30.409

Therefore, 32.35 * 4.94 = 129.4 + 30.409 = 159.809. Which is very close to 160. The difference is 160 - 159.809 = 0.191. Therefore, 0.191 / 32.35 ≈ 0.0059. So total is 4.94 + 0.0059 ≈ 4.9459 USD/hour. Which matches previous result.

So 4.9459 * 8 = ?

Well, 4.9459 * 8. Let me compute 4.9459 * 8:

First, 4.9459 * 2 = 9.8918

Multiply by 4: 9.8918 * 4 = 39.5672. Yes, same as before. So $39.5672.

For Ian: 34 / 6.95 = ?

Let me compute 6.95 * 4.89 = ?

Earlier, we saw that 6.95 * 4.89 ≈ 33.9855. So 34 - 33.9855 = 0.0145. Therefore, 0.0145 / 6.95 ≈ 0.002086. Therefore, total is 4.89 + 0.002086 ≈ 4.892086. Therefore, hourly rate ≈ $4.8921.

Multiply by 8: 4.8921 * 8. Let me compute 4.8921 * 8.

Well, 4.8921 * 8:

Breakdown:

4 * 8 = 32

0.8921 * 8 = ?

0.8 * 8 = 6.4

0.0921 * 8 = 0.7368

Total: 6.4 + 0.7368 = 7.1368

Total earnings: 32 + 7.1368 = 39.1368 USD. So approximately $39.14.

Comparing to Navin's $39.57, Ian is at $39.14. So Navin is higher.

Luka is at $38.61. Therefore, Navin earns the most.

But let me check another way. Let me compute each total earnings using exact formulas without approximating hourly rates. Maybe that would be more accurate?

Let me try that approach.

Total earnings for Navin: 160 rupee/hour * 8 hours = 1280 rupee. Convert to USD: 1280 / 32.35.

Similarly for Luka: 25 kuna/hour * 8 = 200 kuna. Convert to USD: 200 / 5.18.

For Ian: 34 pula/hour * 8 = 272 pula. Convert to USD: 272 / 6.95.

So let me compute each of these.

Starting with Navin: 1280 / 32.35. Let me compute this.

Again, since 32.35 * 40 = 1294. So 1280 is less than that. Let me compute 1280 / 32.35.

Let me note that 32.35 * 39 = ?

Well, 32.35 * 40 = 1294, subtract 32.35: 1294 - 32.35 = 1261.65.

So 32.35 * 39 = 1261.65.

Then 1280 - 1261.65 = 18.35. Therefore, 18.35 / 32.35 ≈ 0.567.

Therefore, total is 39 + 0.567 ≈ 39.567 USD. Which matches the previous value of ~$39.57.

For Luka: 200 / 5.18. Let me compute this.

Again, let me note that 5.18 * 38 = ?

Well, 5 * 38 = 190, 0.18 * 38 = 6.84. Total 190 + 6.84 = 196.84.

So 5.18 * 38 = 196.84

Subtract that from 200: 200 - 196.84 = 3.16.

Then 3.16 / 5.18 ≈ 0.61. Therefore total is 38 + 0.61 ≈ 38.61 USD. Which matches previous value.

For Ian: 272 / 6.95. Let me compute that.

First, compute 6.95 * 39 = ?

Well, 6.95 * 40 = 278. So subtract 6.95: 278 - 6.95 = 271.05.

Therefore, 6.95 * 39 = 271.05.

Subtract that from 272: 272 - 271.05 = 0.95.

Then 0.95 / 6.95 ≈ 0.1367. So total is 39 + 0.1367 ≈ 39.1367 USD. Which is approximately $39.14, as before.

So these exact calculations confirm the same totals:

Navin: ~$39.57

Luka: ~$38.61

Ian: ~$39.14

So Navin still earns the most. Therefore, the answer should be Navin.

But let me check once more if there is any chance I made a mistake in the exchange rates or in the multiplication. Let me verify each step.

First, exchange rates:

- 1 USD = 32.35 Mauritius rupee. So to get rupee to USD, divide by 32.35. Correct.

- 1 USD = 5.18 Croatian kuna. So kuna to USD, divide by 5.18. Correct.

- 1 USD = 6.95 Botswana pula. So pula to USD, divide by 6.95. Correct.

Calculations for Navin:

160 * 8 = 1280 rupee. 1280 / 32.35 ≈ 39.57 USD. Correct.

Calculations for Luka:

25 * 8 = 200 kuna. 200 / 5.18 ≈ 38.61 USD. Correct.

Calculations for Ian:

34 * 8 = 272 pula. 272 / 6.95 ≈ 39.14 USD. Correct.

Yes, seems correct. So Navin earns the most.

But wait, just to be thorough, let me use calculator-like exact decimal computations for each total.

Starting with Navin: 1280 divided by 32.35.

Let me perform this division more accurately.

Let me write 1280 ÷ 32.35.

First, note that 32.35 × 39 = 1261.65 as above.

Subtract that from 1280: 1280 - 1261.65 = 18.35.

Now, 18.35 ÷ 32.35.

Let me compute this fraction. Multiply numerator and denominator by 100: 1835 / 3235. Simplify? Let me divide numerator and denominator by 5: 367 / 647 ≈ ?

Well, 647 × 0.5 = 323.5. 367 - 323.5 = 43.5. So 0.5 + (43.5 / 647).

43.5 / 647 ≈ 0.0672. Therefore, total ≈ 0.5 + 0.0672 = 0.5672.

Therefore, total Navin earnings ≈ 39.5672 USD, as before.

Similarly for Ian: 272 ÷ 6.95.

We had 6.95 × 39 = 271.05.

Then 272 - 271.05 = 0.95.

So 0.95 ÷ 6.95 = ?

Multiply numerator and denominator by 100: 95 / 695. Simplify by dividing numerator and denominator by 5: 19 / 139 ≈ ?

Compute 139 × 0.136 = 139 × 0.1 + 139 × 0.03 + 139 × 0.006 = 13.9 + 4.17 + 0.834 = 13.9 + 4.17 = 18.07 + 0.834 = 18.904. Which is more than 19? Wait, 139 × 0.136 ≈ 18.904. But we need 19. So 0.136 + (19 - 18.904)/139 ≈ 0.136 + 0.096/139 ≈ 0.136 + 0.0007 ≈ 0.1367. Therefore, 0.1367. Hence, total earnings ≈ 39.1367 USD.

So, yes, the exactness is there. Therefore, Navin is ahead of Ian by about $0.43, and ahead of Luka by about $1. So definitely Navin earns the most.

Therefore, the answer is Navin.

But wait, hold on a second. The question says "who would earn the most after working an eight-hour day?" So based on the conversion to USD, Navin is the highest.

But let me check once again if there's any chance I mixed up the exchange rates. For example, maybe the exchange rates are given as USD to local currency, but does that mean that to convert local currency to USD, we have to divide by the rate? Yes. Because if 1 USD is equal to X local currency, then 1 local currency is equal to 1/X USD. So yes, if you have Y local currency, to convert to USD, you divide by X.

For example, Navin's 160 rupee. Since 1 USD = 32.35 rupee, then 1 rupee = 1/32.35 USD. So 160 rupee = 160 * (1/32.35) USD. Which is exactly what I did. So that's correct.

Similarly for others. So calculations are correct.

Therefore, conclusion: Navin earns the most.

But let me check if the question is asking about their earnings in their own currency? No, the question says "who would earn the most after working an eight-hour day?" and given that the exchange rates are provided, it's implied that they want to compare in USD. Because otherwise, if they were comparing in their own currencies, they would have the same rate, but since the exchange rates are given, probably need to convert to a common currency.

Alternatively, perhaps the question is a trick question? Like, maybe the actual earnings in their own currency? But no, because they are chatting in the hostel, and they learn about their wages, but the question is asking who would earn the most, which is likely in USD, since the exchange rates are given.

Alternatively, maybe the question is in the context of their local currencies? But then, without exchange rates, we cannot compare. Since the problem provides exchange rates, it's very likely that we are supposed to convert to USD.

So, I think my conclusion is correct. Navin earns the most.

But just for thoroughness, let me check if there is any possibility that I misread the exchange rates. Let me check again:

"One US dollar is equivalent to 32.35 Mauritius rupee, 5.18 Croatian kuna, and 6.95 Botswana pula."

Therefore, 1 USD = 32.35 rupee => 1 rupee = 1/32.35 USD

Similarly for others. So correct.

Therefore, calculations are correct.

Hence, Navin earns the most.

**Final Answer**
\boxed{Navin}

To determine who earns the most after working an 8-hour day, we must convert each person's hourly wage into **US dollars**, using the given exchange rates. Then, we calculate the total earnings in USD and compare them.

---

### Step 1: Convert Wages to USD

#### Navin (Mauritius):
- Hourly wage: 160 rupee
- Exchange rate: 1 USD = 32.35 rupee
- Hourly wage in USD:
  $$
  \frac{160}{32.35} \approx 4.9459
  $$
- Daily earnings (8 hours):
  $$
  4.9459 \times 8 = 39.5672 \text{ USD}
  $$

#### Luka (Croatia):
- Hourly wage: 25 kuna
- Exchange rate: 1 USD = 5.18 kuna
- Hourly wage in USD:
  $$
  \frac{25}{5.18} \approx 4.8263
  $$
- Daily earnings:
  $$
  4.8263 \times 8 = 38.6104 \text{ USD}
  $$

#### Ian (Botswana):
- Hourly wage: 34 pula
- Exchange rate: 1 USD = 6.95 pula
- Hourly wage in USD:
  $$
  \frac{34}{6.95} \approx 4.8921
  $$
- Daily earnings:
  $$
  4.8921 \times 8 = 39.1368 \text{ USD}
  $$

---

### Step 2: Compare Daily Earnings

| Name   | Daily Earnings (USD) |
|--------|----------------------|
| Navin  | ≈ 39.57              |
| Luka   | ≈ 38.61              |
| Ian    | ≈ 39.14              |

---

### Conclusion

Based on the conversions and calculations, **Navin** earns the most after an 8-hour day, with approximately **$39.57** in US dollars.

$$
\boxed{Navin}
$$

[Assistant]
Explanation:
Okay, let's see. Navin, Luka, and Ian are from different countries, and they each have different hourly wages in their local currencies. The question is asking who would earn the most after working an eight-hour day if we convert their wages to US dollars.

First, let me list out the given information to make sure I have everything straight.

Navin from Mauritius makes 160 rupee per hour. Luka from Croatia makes 25 kuna per hour. Ian from Botswana makes 34 pula per hour.

The exchange rates are:
1 US dollar = 32.35 Mauritius rupee
1 US dollar = 5.18 Croatian kuna
1 US dollar = 6.95 Botswana pula

So, the plan is to convert each of their hourly wages into US dollars, then multiply by 8 hours to get the total earnings for the day, and then compare those totals. The person with the highest total in USD would be the one earning the most.

Let me start with Navin. He makes 160 rupee per hour. Since 1 USD is 32.35 rupee, I need to find out how many USD 160 rupee is. To do that, I can divide his hourly wage by the exchange rate.

So Navin's hourly wage in USD is 160 / 32.35. Let me calculate that.

First, let me write down the calculation: 160 divided by 32.35. Let me get a calculator for this... Wait, maybe I can approximate it? Or is there a better way?

Alternatively, maybe I can compute it step by step. Let me do this division. Let me think. 32.35 times 4 is 129.4. Then 32.35 times 5 is 161.75. Oh, so 32.35 times 5 is 161.75, which is just a bit more than 160. So Navin's wage is slightly less than 5 USD per hour. Let me compute the exact value.

So 160 divided by 32.35. Let me write it as 160 / 32.35. Let me convert this to decimal. Let me multiply numerator and denominator by 100 to eliminate decimals: 16000 / 3235. Let me approximate this division.

Alternatively, since I know that 32.35 * 4.95 = let's see... Well, maybe it's easier to just do decimal division.

Let me try:

32.35 goes into 160 how many times?

Well, 32.35 * 4 = 129.4

Subtract that from 160: 160 - 129.4 = 30.6

Bring down a zero (assuming decimal division). Wait, since it's 160 divided by 32.35, maybe I should write it as 160.00 divided by 32.35.

So 32.35 goes into 160.00 four times (as before), with remainder 30.6. Now, add a decimal point and continue.

So 32.35 goes into 306.0 (after multiplying remainder by 10) how many times? Let's see, 32.35 * 9 = 291.15, which is less than 306. So 9 times. Subtract 291.15 from 306.0: 306.0 - 291.15 = 14.85. Bring down a zero, making it 148.5.

Now, 32.35 goes into 148.5 about 4 times (since 32.35 * 4 = 129.4). Subtract that from 148.5: 148.5 - 129.4 = 19.1. Bring down another zero, making it 191.0.

32.35 goes into 191.0 about 5 times (32.35 * 5 = 161.75). Subtract that: 191.0 - 161.75 = 29.25. Bring down another zero, making it 292.5.

32.35 goes into 292.5 about 9 times (since 32.35 * 9 = 291.15). Subtract that: 292.5 - 291.15 = 1.35.

So putting it all together, we had 4.9 (from the first two steps: 4 and then 9), then 4, then 5, then 9... So approximately 4.9459... So roughly $4.95 per hour.

Wait, let me check that again. Wait, initial division:

First step: 4. Then decimal, then 9, 4, 5, 9... So 4.945... So approximately $4.95 per hour.

But let me check with calculator steps. Let me use another approach. Let me compute 160 / 32.35.

Let me approximate 32.35 as 32.35. Let me note that 32.35 * 5 = 161.75, which is 1.75 more than 160. So, 5 - (1.75 / 32.35). So approximately 5 - 0.054 = 4.946. So yeah, about $4.946 per hour.

So Navin's hourly wage in USD is approximately $4.95.

Now, moving on to Luka. He makes 25 kuna per hour. The exchange rate is 1 USD = 5.18 kuna. Therefore, to find his hourly wage in USD, we divide his wage by 5.18.

So 25 / 5.18. Let me compute this. Let me think.

Again, 5.18 times 4 is 20.72. Subtract that from 25: 25 - 20.72 = 4.28. Then, 5.18 goes into 4.28 about 0.825 times? Let me see.

Alternatively, let me do decimal division. 25 divided by 5.18. Multiply numerator and denominator by 100 to get rid of decimals: 2500 / 518. Let me approximate this.

518 * 4 = 2072. 2500 - 2072 = 428. Bring down a zero (making it 4280). 518 * 8 = 4144. Subtract from 4280: 4280 - 4144 = 136. Bring down another zero, making it 1360. 518 * 2 = 1036. Subtract: 1360 - 1036 = 324. Bring down another zero, making it 3240. 518 * 6 = 3108. Subtract: 3240 - 3108 = 132.

So so far, we have 4.826... So approximately $4.83 per hour. Let me check with calculator steps. Since 5.18 * 4.8 = let's compute that. 5 * 4.8 = 24, 0.18 * 4.8 = 0.864. Total is 24.864. Which is less than 25. The difference is 25 - 24.864 = 0.136. So then 0.136 / 5.18 ≈ 0.0263. So total is approximately 4.8 + 0.0263 ≈ 4.8263. So yeah, approximately $4.83 per hour. So Luka's hourly wage is roughly $4.83.

Now, Ian from Botswana makes 34 pula per hour. The exchange rate is 1 USD = 6.95 pula. Therefore, his hourly wage in USD is 34 divided by 6.95. Let me compute that.

Again, let me do decimal division. 34 / 6.95. Multiply numerator and denominator by 100: 3400 / 695. Let me simplify that.

First, note that 695 * 4 = 2780. Subtract that from 3400: 3400 - 2780 = 620. Bring down a zero (making it 6200). 695 * 8 = 5560. Subtract: 6200 - 5560 = 640. Bring down another zero, making it 6400. 695 * 9 = 6255. Subtract: 6400 - 6255 = 145. Bring down another zero, making it 1450. 695 * 2 = 1390. Subtract: 1450 - 1390 = 60.

So so far, we have 4.892... So approximately $4.89 per hour. Let me check with another method. Let's see, 6.95 * 4.8 = ?

First, 6.95 * 4 = 27.8, 6.95 * 0.8 = 5.56. Total is 27.8 + 5.56 = 33.36. Which is less than 34. The difference is 34 - 33.36 = 0.64. Then 0.64 / 6.95 ≈ 0.092. So total is 4.8 + 0.092 ≈ 4.892. So yes, approximately $4.89 per hour. So Ian's hourly wage is roughly $4.89 per hour.

So summarizing:

Navin: ~$4.95 per hour

Luka: ~$4.83 per hour

Ian: ~$4.89 per hour

So Navin has the highest hourly wage in USD. Therefore, after working 8 hours, Navin would earn the most. But let me check again to make sure I didn't make a mistake in calculations.

Wait, but let me verify the exact numbers again. Maybe my approximations are off? Let me compute each one more precisely.

Starting with Navin: 160 / 32.35. Let me use a calculator-like approach. Let me note that 32.35 * 4.9 = ?

Well, 32.35 * 4 = 129.4

32.35 * 0.9 = 29.115

Adding those gives 129.4 + 29.115 = 158.515. Which is less than 160. The difference is 160 - 158.515 = 1.485. So 1.485 / 32.35 ≈ 0.0459. Therefore, total is 4.9 + 0.0459 ≈ 4.9459. So $4.9459 per hour.

So Navin's hourly rate is approximately $4.9459.

For Luka: 25 / 5.18. Let me compute that. Let me consider that 5.18 * 4.8 = 24.864 (as before). The difference between 25 and 24.864 is 0.136. Then 0.136 / 5.18 ≈ 0.0263. So total is 4.8 + 0.0263 ≈ 4.8263. So approximately $4.8263 per hour.

For Ian: 34 / 6.95. Let me compute 6.95 * 4.89 = ?

Well, 6.95 * 4 = 27.8

6.95 * 0.89 = ?

First, 6.95 * 0.8 = 5.56

6.95 * 0.09 = 0.6255

Adding those gives 5.56 + 0.6255 = 6.1855

So total is 27.8 + 6.1855 = 33.9855, which is very close to 34. The difference is 34 - 33.9855 = 0.0145. Therefore, 0.0145 / 6.95 ≈ 0.002086. Therefore, total rate is approximately 4.89 + 0.002086 ≈ 4.8921. So approximately $4.8921 per hour.

Therefore, precise rates:

Navin: ~$4.9459

Luka: ~$4.8263

Ian: ~$4.8921

So Navin still has the highest hourly rate. Therefore, over 8 hours, their earnings would be:

Navin: 4.9459 * 8

Luka: 4.8263 * 8

Ian: 4.8921 * 8

Let me compute each of those.

Starting with Navin:

4.9459 * 8. Let me compute 4.9459 * 8.

Well, 4 * 8 = 32

0.9459 * 8 = ?

0.9 * 8 = 7.2

0.0459 * 8 = 0.3672

So total is 7.2 + 0.3672 = 7.5672

Therefore, total earnings: 32 + 7.5672 = 39.5672 USD. Approximately $39.57.

Luka: 4.8263 * 8.

Let me compute that. 4 * 8 = 32. 0.8263 * 8 = ?

0.8 * 8 = 6.4

0.0263 * 8 = 0.2104

Total: 6.4 + 0.2104 = 6.6104

So total earnings: 32 + 6.6104 = 38.6104 USD. Approximately $38.61.

Ian: 4.8921 * 8.

Compute 4 * 8 = 32. 0.8921 * 8.

0.8 * 8 = 6.4

0.0921 * 8 = 0.7368

Total: 6.4 + 0.7368 = 7.1368

Therefore, total earnings: 32 + 7.1368 = 39.1368 USD. Approximately $39.14.

So Navin earns approximately $39.57, Luka $38.61, Ian $39.14. Therefore, Navin earns the most.

But wait, let me check my calculations again to make sure.

For Navin: 160 rupee per hour. 1 USD = 32.35 rupee. So 160 / 32.35. Let me compute 160 / 32.35 with more precision. Let me do 160 / 32.35.

Let me note that 32.35 * 4.94 = ?

Let me compute 32.35 * 4 = 129.4

32.35 * 0.94 = ?

First, 32.35 * 0.9 = 29.115

32.35 * 0.04 = 1.294

Adding them: 29.115 + 1.294 = 30.409

Therefore, 32.35 * 4.94 = 129.4 + 30.409 = 159.809. Which is very close to 160. The difference is 160 - 159.809 = 0.191. Therefore, 0.191 / 32.35 ≈ 0.0059. So total is 4.94 + 0.0059 ≈ 4.9459 USD/hour. Which matches previous result.

So 4.9459 * 8 = ?

Well, 4.9459 * 8. Let me compute 4.9459 * 8:

First, 4.9459 * 2 = 9.8918

Multiply by 4: 9.8918 * 4 = 39.5672. Yes, same as before. So $39.5672.

For Ian: 34 / 6.95 = ?

Let me compute 6.95 * 4.89 = ?

Earlier, we saw that 6.95 * 4.89 ≈ 33.9855. So 34 - 33.9855 = 0.0145. Therefore, 0.0145 / 6.95 ≈ 0.002086. Therefore, total is 4.89 + 0.002086 ≈ 4.892086. Therefore, hourly rate ≈ $4.8921.

Multiply by 8: 4.8921 * 8. Let me compute 4.8921 * 8.

Well, 4.8921 * 8:

Breakdown:

4 * 8 = 32

0.8921 * 8 = ?

0.8 * 8 = 6.4

0.0921 * 8 = 0.7368

Total: 6.4 + 0.7368 = 7.1368

Total earnings: 32 + 7.1368 = 39.1368 USD. So approximately $39.14.

Comparing to Navin's $39.57, Ian is at $39.14. So Navin is higher.

Luka is at $38.61. Therefore, Navin earns the most.

But let me check another way. Let me compute each total earnings using exact formulas without approximating hourly rates. Maybe that would be more accurate?

Let me try that approach.

Total earnings for Navin: 160 rupee/hour * 8 hours = 1280 rupee. Convert to USD: 1280 / 32.35.

Similarly for Luka: 25 kuna/hour * 8 = 200 kuna. Convert to USD: 200 / 5.18.

For Ian: 34 pula/hour * 8 = 272 pula. Convert to USD: 272 / 6.95.

So let me compute each of these.

Starting with Navin: 1280 / 32.35. Let me compute this.

Again, since 32.35 * 40 = 1294. So 1280 is less than that. Let me compute 1280 / 32.35.

Let me note that 32.35 * 39 = ?

Well, 32.35 * 40 = 1294, subtract 32.35: 1294 - 32.35 = 1261.65.

So 32.35 * 39 = 1261.65.

Then 1280 - 1261.65 = 18.35. Therefore, 18.35 / 32.35 ≈ 0.567.

Therefore, total is 39 + 0.567 ≈ 39.567 USD. Which matches the previous value of ~$39.57.

For Luka: 200 / 5.18. Let me compute this.

Again, let me note that 5.18 * 38 = ?

Well, 5 * 38 = 190, 0.18 * 38 = 6.84. Total 190 + 6.84 = 196.84.

So 5.18 * 38 = 196.84

Subtract that from 200: 200 - 196.84 = 3.16.

Then 3.16 / 5.18 ≈ 0.61. Therefore total is 38 + 0.61 ≈ 38.61 USD. Which matches previous value.

For Ian: 272 / 6.95. Let me compute that.

First, compute 6.95 * 39 = ?

Well, 6.95 * 40 = 278. So subtract 6.95: 278 - 6.95 = 271.05.

Therefore, 6.95 * 39 = 271.05.

Subtract that from 272: 272 - 271.05 = 0.95.

Then 0.95 / 6.95 ≈ 0.1367. So total is 39 + 0.1367 ≈ 39.1367 USD. Which is approximately $39.14, as before.

So these exact calculations confirm the same totals:

Navin: ~$39.57

Luka: ~$38.61

Ian: ~$39.14

So Navin still earns the most. Therefore, the answer should be Navin.

But let me check once more if there is any chance I made a mistake in the exchange rates or in the multiplication. Let me verify each step.

First, exchange rates:

- 1 USD = 32.35 Mauritius rupee. So to get rupee to USD, divide by 32.35. Correct.

- 1 USD = 5.18 Croatian kuna. So kuna to USD, divide by 5.18. Correct.

- 1 USD = 6.95 Botswana pula. So pula to USD, divide by 6.95. Correct.

Calculations for Navin:

160 * 8 = 1280 rupee. 1280 / 32.35 ≈ 39.57 USD. Correct.

Calculations for Luka:

25 * 8 = 200 kuna. 200 / 5.18 ≈ 38.61 USD. Correct.

Calculations for Ian:

34 * 8 = 272 pula. 272 / 6.95 ≈ 39.14 USD. Correct.

Yes, seems correct. So Navin earns the most.

But wait, just to be thorough, let me use calculator-like exact decimal computations for each total.

Starting with Navin: 1280 divided by 32.35.

Let me perform this division more accurately.

Let me write 1280 ÷ 32.35.

First, note that 32.35 × 39 = 1261.65 as above.

Subtract that from 1280: 1280 - 1261.65 = 18.35.

Now, 18.35 ÷ 32.35.

Let me compute this fraction. Multiply numerator and denominator by 100: 1835 / 3235. Simplify? Let me divide numerator and denominator by 5: 367 / 647 ≈ ?

Well, 647 × 0.5 = 323.5. 367 - 323.5 = 43.5. So 0.5 + (43.5 / 647).

43.5 / 647 ≈ 0.0672. Therefore, total ≈ 0.5 + 0.0672 = 0.5672.

Therefore, total Navin earnings ≈ 39.5672 USD, as before.

Similarly for Ian: 272 ÷ 6.95.

We had 6.95 × 39 = 271.05.

Then 272 - 271.05 = 0.95.

So 0.95 ÷ 6.95 = ?

Multiply numerator and denominator by 100: 95 / 695. Simplify by dividing numerator and denominator by 5: 19 / 139 ≈ ?

Compute 139 × 0.136 = 139 × 0.1 + 139 × 0.03 + 139 × 0.006 = 13.9 + 4.17 + 0.834 = 13.9 + 4.17 = 18.07 + 0.834 = 18.904. Which is more than 19? Wait, 139 × 0.136 ≈ 18.904. But we need 19. So 0.136 + (19 - 18.904)/139 ≈ 0.136 + 0.096/139 ≈ 0.136 + 0.0007 ≈ 0.1367. Therefore, 0.1367. Hence, total earnings ≈ 39.1367 USD.

So, yes, the exactness is there. Therefore, Navin is ahead of Ian by about $0.43, and ahead of Luka by about $1. So definitely Navin earns the most.

Therefore, the answer is Navin.

But wait, hold on a second. The question says "who would earn the most after working an eight-hour day?" So based on the conversion to USD, Navin is the highest.

But let me check once again if there's any chance I mixed up the exchange rates. For example, maybe the exchange rates are given as USD to local currency, but does that mean that to convert local currency to USD, we have to divide by the rate? Yes. Because if 1 USD is equal to X local currency, then 1 local currency is equal to 1/X USD. So yes, if you have Y local currency, to convert to USD, you divide by X.

For example, Navin's 160 rupee. Since 1 USD = 32.35 rupee, then 1 rupee = 1/32.35 USD. So 160 rupee = 160 * (1/32.35) USD. Which is exactly what I did. So that's correct.

Similarly for others. So calculations are correct.

Therefore, conclusion: Navin earns the most.

But let me check if the question is asking about their earnings in their own currency? No, the question says "who would earn the most after working an eight-hour day?" and given that the exchange rates are provided, it's implied that they want to compare in USD. Because otherwise, if they were comparing in their own currencies, they would have the same rate, but since the exchange rates are given, probably need to convert to a common currency.

Alternatively, perhaps the question is a trick question? Like, maybe the actual earnings in their own currency? But no, because they are chatting in the hostel, and they learn about their wages, but the question is asking who would earn the most, which is likely in USD, since the exchange rates are given.

Alternatively, maybe the question is in the context of their local currencies? But then, without exchange rates, we cannot compare. Since the problem provides exchange rates, it's very likely that we are supposed to convert to USD.

So, I think my conclusion is correct. Navin earns the most.

But just for thoroughness, let me check if there is any possibility that I misread the exchange rates. Let me check again:

"One US dollar is equivalent to 32.35 Mauritius rupee, 5.18 Croatian kuna, and 6.95 Botswana pula."

Therefore, 1 USD = 32.35 rupee => 1 rupee = 1/32.35 USD

Similarly for others. So correct.

Therefore, calculations are correct.

Hence, Navin earns the most.

**Final Answer**
\boxed{Navin}

To determine who earns the most after working an 8-hour day, we must convert each person's hourly wage into **US dollars**, using the given exchange rates. Then, we calculate the total earnings in USD and compare them.

---

### Step 1: Convert Wages to USD

#### Navin (Mauritius):
- Hourly wage: 160 rupee
- Exchange rate: 1 USD = 32.35 rupee
- Hourly wage in USD:
  $$
  \frac{160}{32.35} \approx 4.9459
  $$
- Daily earnings (8 hours):
  $$
  4.9459 \times 8 = 39.5672 \text{ USD}
  $$

#### Luka (Croatia):
- Hourly wage: 25 kuna
- Exchange rate: 1 USD = 5.18 kuna
- Hourly wage in USD:
  $$
  \frac{25}{5.18} \approx 4.8263
  $$
- Daily earnings:
  $$
  4.8263 \times 8 = 38.6104 \text{ USD}
  $$

#### Ian (Botswana):
- Hourly wage: 34 pula
- Exchange rate: 1 USD = 6.95 pula
- Hourly wage in USD:
  $$
  \frac{34}{6.95} \approx 4.8921
  $$
- Daily earnings:
  $$
  4.8921 \times 8 = 39.1368 \text{ USD}
  $$

---

### Step 2: Compare Daily Earnings

| Name   | Daily Earnings (USD) |
|--------|----------------------|
| Navin  | ≈ 39.57              |
| Luka   | ≈ 38.61              |
| Ian    | ≈ 39.14              |

---

### Conclusion

Based on the conversions and calculations, **Navin** earns the most after an 8-hour day, with approximately **$39.57** in US dollars.

$$
\boxed{Navin}
$$

Solution: The easiest way to solve this problem is to convert the wages to US dollars and to ignore the eight-hour day. In one hour Navin makes $160\text{ rupee} \times \frac{1\text{ USD}}{32.35\text{ rupee}}\approx 4.95\; \text{USD}$. Luka makes $25 \text{ kuna} \times \frac{1\text{ USD}}{5.18 \text{ kuna}}\approx 4.83 \text{ USD}$. Ian makes $34\text{ pula} \times\frac{1\text{ USD}}{6.95 \text{ pula}}\approx 4.89 \text{ USD}$. Comparing these numbers, we see that $\boxed{\text{Navin}}$ has the highest per-hour wages, and therefore would earn the most in eight hours.