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 expression is always non-negative. But here, the entire fraction is less than zero, which means the numerator and denominator must have opposite signs. However, since the numerator is an absolute value, it's always non-negative. So, the numerator is either zero or positive. Therefore, for the whole fraction to be negative, the denominator must be negative. But wait, let me check that again.

The absolute value $|x^2 - 81|$ is always greater than or equal to zero. So, the numerator is non-negative. The denominator is $x^2 - 36x$. For the entire fraction to be less than zero, the denominator must be negative because a non-negative number divided by a negative number is negative. However, if the numerator is zero, then the entire fraction is zero, which is not less than zero. Therefore, the inequality will hold when the denominator is negative and the numerator is not zero. So, my first step is to find where the denominator is negative and then exclude the points where the numerator is zero.

But let me write that down step by step.

First, let me note that the expression is undefined when the denominator is zero. So, I need to find the domain first. The denominator $x^2 - 36x$ factors as $x(x - 36)$. Therefore, the denominator is zero when $x = 0$ or $x = 36$. Therefore, the domain of the expression is all real numbers except $x = 0$ and $x = 36$.

Now, the inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. As I thought earlier, since the numerator is non-negative, the fraction is negative only when the denominator is negative. Therefore, I need to find the values of x where $x^2 - 36x < 0$ and $|x^2 - 81| \neq 0$.

So, first, let me solve $x^2 - 36x < 0$. Let me factor that:

$x(x - 36) < 0$.

To solve this inequality, I can use a sign chart. The critical points are at x = 0 and x = 36. These divide the number line into three intervals:

1. $x < 0$
2. $0 < x < 36$
3. $x > 36$

Testing each interval:

1. For $x < 0$, let's pick x = -1. Then, (-1)(-1 - 36) = (-1)(-37) = 37 > 0. So, positive.
2. For $0 < x < 36$, let's pick x = 18. Then, 18(18 - 36) = 18(-18) = -324 < 0. So, negative.
3. For $x > 36$, let's pick x = 40. Then, 40(40 - 36) = 40(4) = 160 > 0. So, positive.

Therefore, the inequality $x^2 - 36x < 0$ holds when $0 < x < 36$.

But remember, we also need to exclude the points where the numerator is zero because if the numerator is zero, the entire expression is zero, which is not less than zero. So, the numerator $|x^2 - 81|$ is zero when $x^2 - 81 = 0$, which is when $x^2 = 81$, so $x = 9$ or $x = -9$.

Therefore, in the interval $0 < x < 36$, we need to exclude x = 9 because at x = 9, the numerator is zero, making the entire expression zero, which does not satisfy the strict inequality.

So, putting it all together, the solution set is $0 < x < 36$ excluding x = 9. However, we also need to check if there are any other restrictions. Wait, but the original domain excludes x = 0 and x = 36, which are already excluded in the interval $0 < x < 36$. So, the solution is $0 < x < 36$ with x ≠ 9.

But let me verify this. Let me check if there are any other points where the numerator is zero. The numerator is |x² - 81|, which is zero when x² - 81 = 0, so x = ±9. So, in the interval 0 < x < 36, x = 9 is the only one. x = -9 is not in this interval, so we don't have to worry about that. Therefore, the solution is (0, 9) ∪ (9, 36).

But wait, let me check if there are any other points where the denominator is zero. The denominator is zero at x = 0 and x = 36, which are already excluded. So, the answer is all real numbers between 0 and 36, not including 0, 36, and 9.

But let me test some values to make sure. Let's pick x = 1, which is in (0, 9). Then, numerator is |1 - 81| = | -80 | = 80. Denominator is 1 - 36 = -35. So, 80 / (-35) = -80/35 < 0. So, that works.

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

Now, x = 9. The numerator is zero, so the expression is zero, which is not less than zero. So, x = 9 is excluded. Similarly, x = 0 and x = 36 are excluded because denominator is zero.

What about x = -1? Wait, x = -1 is not in the interval (0, 36), so it's not part of the solution. Let me check. If x = -1, denominator is (-1)^2 - 36*(-1) = 1 + 36 = 37 > 0. So, the fraction is |(-1)^2 - 81| / 37 = |1 - 81| / 37 = 80 / 37 > 0, which is not less than zero. So, that's correct.

But wait, what if x is negative? The denominator is positive for x < 0, so the fraction is non-negative (since numerator is non-negative) divided by positive, so non-negative. Therefore, the inequality is not satisfied for x < 0. Similarly, for x > 36, denominator is positive, so same thing. Therefore, only between 0 and 36, denominator is negative, so the fraction is negative, except when numerator is zero. Therefore, the solution is indeed (0, 9) ∪ (9, 36).

But let me check another point in (0, 9). Let's take x = 5. Numerator: |25 - 81| = 56. Denominator: 25 - 180 = -155. So, 56 / (-155) < 0. Correct.

Another point in (9, 36): x = 18. Numerator: |324 - 81| = 243. Denominator: 324 - 648 = -324. So, 243 / (-324) = -0.75 < 0. Correct.

What about x approaching 0 from the right? Let's say x approaches 0+. Then denominator approaches 0 - 0 = 0? Wait, denominator is x² - 36x. As x approaches 0+, x² approaches 0, and -36x approaches 0 from the negative side. So, denominator approaches 0 from the negative side. The numerator approaches |0 - 81| = 81. Therefore, the expression approaches 81 / 0- which is negative infinity. So, it's negative near x = 0. Similarly, approaching x = 36 from the left, denominator approaches 0 from the negative side. Let me check. Let x approach 36 from the left, so x = 36 - ε, where ε is small positive. Then denominator is (36 - ε)^2 - 36*(36 - ε) = 36² - 72ε + ε² - 36*36 + 36ε = (-72ε + 36ε) + ε² = -36ε + ε² ≈ -36ε < 0. So denominator approaches 0 from the negative side. The numerator is |(36 - ε)^2 - 81|. Let me compute (36 - ε)^2 - 81 = 1296 - 72ε + ε² - 81 = 1215 - 72ε + ε². Since ε is small, this is approximately 1215, which is positive. Therefore, the absolute value is 1215 - 72ε + ε², so the numerator approaches 1215. Therefore, the expression approaches 1215 / 0- which is negative infinity. So, near x = 36, the expression is also negative. Therefore, the entire interval (0, 36) except x = 9 is the solution.

But let me check if there are any other points where the numerator is zero. For example, x = -9. If x = -9, then the numerator is zero, but the denominator is (-9)^2 - 36*(-9) = 81 + 324 = 405 > 0. Therefore, the expression is 0 / 405 = 0, which is not less than zero. Therefore, x = -9 is not part of the solution. So, only x = 9 is excluded in the interval (0, 36).

Therefore, the solution set is all real numbers between 0 and 36, not including 0, 9, and 36. So, in interval notation, that's (0, 9) ∪ (9, 36).

But let me check if there's a possibility that the absolute value could affect the sign in some way. Wait, the absolute value is always non-negative, so the numerator is non-negative. The denominator is negative in (0, 36). Therefore, the entire expression is non-positive (since non-negative divided by negative is non-positive). But the inequality is strict: less than zero. Therefore, we need to exclude the points where the numerator is zero, which is x = ±9. But in the interval (0, 36), only x = 9 is in there. Therefore, the solution is indeed (0, 9) ∪ (9, 36).

Wait, but let me think again. Suppose that the numerator is zero. Then the expression is zero, which is not less than zero. Therefore, we need to exclude x = 9. So, yes, that's correct.

But let me also check if there are any other points where the expression could be undefined. The denominator is zero at x = 0 and x = 36, so those are excluded. So, the answer is (0, 9) ∪ (9, 36).

But let me check if there's a case where the absolute value could make the numerator positive or negative? Wait, no. The absolute value is always non-negative. So, regardless of the value inside, the absolute value is non-negative. Therefore, the numerator is non-negative. Therefore, the sign of the entire expression depends only on the denominator. Therefore, the expression is negative when denominator is negative and numerator is positive, or denominator is negative and numerator is zero? Wait, if numerator is zero, then the expression is zero. Therefore, the inequality is strict, so we need the expression to be negative, which requires denominator negative and numerator positive. Therefore, we need to exclude the points where numerator is zero. Therefore, the solution is denominator negative (which is 0 < x < 36) and numerator not zero (so x ≠ 9). Therefore, the solution is (0, 9) ∪ (9, 36). So, that seems correct.

But let me think again. Suppose that the numerator is zero. Then, the expression is zero. So, the inequality is not satisfied. Therefore, we need to exclude x = 9. Therefore, the answer is (0, 9) ∪ (9, 36). Therefore, I think that is the solution.

But let me check if there's a case where the absolute value could be negative? No, absolute value is always non-negative. Therefore, the numerator is non-negative. Therefore, the fraction is negative only when denominator is negative and numerator is positive. But since numerator is non-negative, the only way for the fraction to be negative is denominator negative and numerator positive. However, if numerator is zero, then the fraction is zero. Therefore, the solution is denominator negative and numerator positive. Therefore, we need to find x where denominator is negative (so 0 < x < 36) and numerator is positive (so |x² - 81| > 0). Which is equivalent to x² - 81 ≠ 0, so x ≠ ±9. Therefore, in the interval 0 < x < 36, x ≠ 9. Therefore, the solution is (0, 9) ∪ (9, 36). Therefore, that's the answer.

But let me check if there's a case where even though the denominator is negative, the numerator is zero. Which is only at x = 9. Therefore, that's the only point to exclude. Therefore, the answer is correct.

Alternatively, maybe I can solve the inequality by considering cases based on the absolute value. Let me try that approach as a check.

The absolute value |x² - 81| can be written as:

|x² - 81| = { x² - 81, if x² - 81 ≥ 0; -(x² - 81), if x² - 81 < 0 }

Which simplifies to:

|x² - 81| = { x² - 81, if |x| ≥ 9; 81 - x², if |x| < 9 }

Therefore, the original inequality is:

Case 1: |x| ≥ 9, so |x² - 81| = x² - 81. Then, the inequality becomes (x² - 81)/(x² - 36x) < 0.

Case 2: |x| < 9, so |x² - 81| = 81 - x². Then, the inequality becomes (81 - x²)/(x² - 36x) < 0.

So, let me analyze both cases.

First, note that the original domain excludes x = 0 and x = 36. Also, in Case 1, |x| ≥ 9, so x ≤ -9 or x ≥ 9. In Case 2, |x| < 9, so -9 < x < 9.

But let me handle each case separately.

Case 1: |x| ≥ 9, so x ≤ -9 or x ≥ 9. The inequality is (x² - 81)/(x² - 36x) < 0.

Let me simplify the expression. Let me factor numerator and denominator.

Numerator: x² - 81 = (x - 9)(x + 9).

Denominator: x² - 36x = x(x - 36).

Therefore, the expression becomes [(x - 9)(x + 9)] / [x(x - 36)] < 0.

So, we need to find where this fraction is negative. Let me note that in Case 1, x ≤ -9 or x ≥ 9. Let me consider these two subcases.

Subcase 1a: x ≤ -9.

In this case, x is negative, and x ≤ -9. Let me check the sign of each factor:

(x - 9): x ≤ -9, so x - 9 ≤ -18 < 0.

(x + 9): x ≤ -9, so x + 9 ≤ 0. But since x ≤ -9, x + 9 ≤ 0. However, when x = -9, x + 9 = 0. So, for x < -9, x + 9 < 0.

x: x ≤ -9, so x < 0.

(x - 36): x ≤ -9, so x - 36 ≤ -45 < 0.

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

Denominator: (negative)(negative) = positive.

Therefore, the entire expression is positive / positive = positive. Therefore, the inequality (positive) < 0 is false. Therefore, no solutions in this subcase.

Subcase 1b: x ≥ 9.

In this case, x ≥ 9. Let me check the sign of each factor:

(x - 9): x ≥ 9, so x - 9 ≥ 0. At x = 9, it's zero.

(x + 9): x ≥ 9, so x + 9 ≥ 18 > 0.

x: x ≥ 9, so x > 0 (since x ≥ 9).

(x - 36): x ≥ 9. So, when x < 36, x - 36 < 0. When x = 36, it's zero. When x > 36, it's positive.

Therefore, the expression [(x - 9)(x + 9)] / [x(x - 36)].

Let me split into intervals:

First, x ≥ 9. The critical points are x = 9, x = 36.

So, intervals:

1. 9 < x < 36

2. x = 9

3. x = 36

4. x > 36

But since we are in Case 1 (x ≥ 9), let's check each interval.

Interval 1: 9 < x < 36.

(x - 9): positive

(x + 9): positive

x: positive

(x - 36): negative

Therefore, numerator: positive * positive = positive

Denominator: positive * negative = negative

Therefore, the expression is positive / negative = negative. Therefore, the inequality holds here. So, 9 < x < 36 is part of the solution.

At x = 9: numerator is zero, so expression is zero. Not less than zero. Excluded.

At x = 36: denominator is zero. Excluded.

Interval 4: x > 36.

(x - 9): positive

(x + 9): positive

x: positive

(x - 36): positive

Therefore, numerator: positive * positive = positive

Denominator: positive * positive = positive

Expression: positive / positive = positive. Not less than zero. Therefore, no solution here.

Therefore, in Case 1, the solution is 9 < x < 36.

Case 2: |x| < 9, so -9 < x < 9. The inequality is (81 - x²)/(x² - 36x) < 0.

Again, let me factor numerator and denominator.

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

Denominator: x² - 36x = x(x - 36).

Therefore, the expression becomes [ - (x - 9)(x + 9) ] / [x(x - 36)] < 0.

Which is equivalent to - (x - 9)(x + 9) / [x(x - 36)] < 0.

Multiply both sides by -1 (remembering to reverse the inequality sign):

(x - 9)(x + 9) / [x(x - 36)] > 0.

So, now we need to solve (x - 9)(x + 9) / [x(x - 36)] > 0.

But in Case 2, we have -9 < x < 9. Let me note that in this interval, x is between -9 and 9, so x ≠ 0 (since denominator is zero at x = 0, which is excluded). Also, x ≠ 9 or -9, but since we are in -9 < x < 9, x ≠ -9 and x ≠ 9 are already excluded.

So, let me analyze the sign of the expression (x - 9)(x + 9) / [x(x - 36)] in the interval -9 < x < 9.

First, let me note that in this interval:

x - 9: since x < 9, x - 9 < 0.

x + 9: since x > -9, x + 9 > 0.

x: in -9 < x < 9, x can be positive or negative. Specifically, x is in (-9, 0) or (0, 9).

x - 36: since x < 9 < 36, x - 36 < 0.

Therefore, let me write the sign of each factor:

(x - 9): negative

(x + 9): positive

x: depends on the subinterval

(x - 36): negative

Therefore, the numerator (x - 9)(x + 9) is negative * positive = negative.

Denominator x(x - 36): x * negative.

So, denominator is x * negative. Let me consider two subcases:

Subcase 2a: -9 < x < 0.

In this case, x is negative. Therefore, denominator: x (negative) * (negative) = positive.

Therefore, the entire expression is numerator (negative) / denominator (positive) = negative. Therefore, the inequality (negative) > 0 is false. Therefore, no solution here.

Subcase 2b: 0 < x < 9.

In this case, x is positive. Therefore, denominator: x (positive) * (negative) = negative.

Therefore, the entire expression is numerator (negative) / denominator (negative) = positive. Therefore, the inequality (positive) > 0 is true. Therefore, this interval is part of the solution.

Therefore, in Case 2, the solution is 0 < x < 9.

Therefore, combining Case 1 and Case 2, the solutions are:

From Case 1: 9 < x < 36

From Case 2: 0 < x < 9

Therefore, the total solution is (0, 9) ∪ (9, 36), which matches my previous conclusion.

Therefore, the answer is all real numbers between 0 and 36, excluding 0, 9, and 36. Therefore, in interval notation, that is (0, 9) ∪ (9, 36).

But let me check if there are any other points where the expression could be negative. For example, in Case 2, when we had 0 < x < 9, the expression was positive? Wait, no. Wait, in Case 2, after multiplying by -1, we had the inequality (x - 9)(x + 9)/[x(x - 36)] > 0. Then, in Subcase 2b, 0 < x < 9, the expression was positive, so the original inequality (before multiplying by -1) was negative. Wait, let me retrace.

Original Case 2 inequality: (81 - x²)/(x² - 36x) < 0.

Which became - (x - 9)(x + 9)/[x(x - 36)] < 0.

Then, multiplying both sides by -1 reverses the inequality:

(x - 9)(x + 9)/[x(x - 36)] > 0.

So, solving this inequality gives us the solution for Case 2. In Subcase 2b, 0 < x < 9, we found that the expression (x - 9)(x + 9)/[x(x - 36)] is positive, so the original inequality holds. Therefore, in Case 2, 0 < x < 9 is part of the solution. Therefore, combining with Case 1, the total solution is (0, 9) ∪ (9, 36). Therefore, that's correct.

Therefore, after checking both cases, I confirm that the solution is (0, 9) ∪ (9, 36).

But let me check if there's any mistake in the sign analysis. For example, in Case 2, when I had 0 < x < 9, the expression (x - 9)(x + 9)/[x(x - 36)]:

(x - 9) is negative, (x + 9) is positive, x is positive, (x - 36) is negative. Therefore, numerator: negative * positive = negative. Denominator: positive * negative = negative. Therefore, negative / negative = positive. Therefore, the inequality (positive) > 0 is true. Therefore, original inequality holds. Therefore, correct.

Therefore, the answer is indeed (0, 9) ∪ (9, 36).

Therefore, I think this is the correct solution.

**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 proceed step by step, analyzing the behavior of the numerator and denominator.

---

### **Step 1: Domain Considerations**

The expression is undefined when the denominator is zero.
$$
x^2 - 36x = x(x - 36) = 0 \Rightarrow x = 0 \text{ or } x = 36.
$$
So, the domain of the expression is all real numbers **except** $ x = 0 $ and $ x = 36 $.

---

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

The numerator is an absolute value:
$$
|x^2 - 81| \geq 0 \quad \text{for all } x.
$$
So, the numerator is **non-negative**.
The inequality is strict:
$$
\frac{|x^2 - 81|}{x^2 - 36x} < 0.
$$
This means the **denominator must be negative**, and the **numerator must not be zero** (since a zero numerator would make the expression zero, not negative).

So, we need to find the values of $ x $ such that:
- $ x^2 - 36x < 0 $, and
- $ |x^2 - 81| \neq 0 $.

---

### **Step 3: Solve the Inequality $ x^2 - 36x < 0 $**

Factor the denominator:
$$
x^2 - 36x = x(x - 36).
$$
This is a quadratic that opens upwards, with roots at $ x = 0 $ and $ x = 36 $.
Using a sign chart or test points, we find:
- $ x < 0 $: positive
- $ 0 < x < 36 $: negative
- $ x > 36 $: positive

So, the inequality $ x^2 - 36x < 0 $ holds for:
$$
0 < x < 36.
$$

---

### **Step 4: Exclude Points Where the Numerator is Zero**

The numerator is zero when:
$$
|x^2 - 81| = 0 \Rightarrow x^2 - 81 = 0 \Rightarrow x = \pm 9.
$$
So, we must **exclude** $ x = 9 $ from the interval $ (0, 36) $, because at $ x = 9 $, the expression becomes zero, which does **not** satisfy the strict inequality.

---

### **Step 5: Final Solution**

Combining the above:
- The expression is negative when $ 0 < x < 36 $,
- But we must exclude $ x = 9 $ (where the numerator is zero),
- And we must also exclude $ x = 0 $ and $ x = 36 $ (where the expression is undefined).

Thus, the solution set is:
$$
(0, 9) \cup (9, 36).
$$

---

### **Final Answer**

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

[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 expression is always non-negative. But here, the entire fraction is less than zero, which means the numerator and denominator must have opposite signs. However, since the numerator is an absolute value, it's always non-negative. So, the numerator is either zero or positive. Therefore, for the whole fraction to be negative, the denominator must be negative. But wait, let me check that again.

The absolute value $|x^2 - 81|$ is always greater than or equal to zero. So, the numerator is non-negative. The denominator is $x^2 - 36x$. For the entire fraction to be less than zero, the denominator must be negative because a non-negative number divided by a negative number is negative. However, if the numerator is zero, then the entire fraction is zero, which is not less than zero. Therefore, the inequality will hold when the denominator is negative and the numerator is not zero. So, my first step is to find where the denominator is negative and then exclude the points where the numerator is zero.

But let me write that down step by step.

First, let me note that the expression is undefined when the denominator is zero. So, I need to find the domain first. The denominator $x^2 - 36x$ factors as $x(x - 36)$. Therefore, the denominator is zero when $x = 0$ or $x = 36$. Therefore, the domain of the expression is all real numbers except $x = 0$ and $x = 36$.

Now, the inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. As I thought earlier, since the numerator is non-negative, the fraction is negative only when the denominator is negative. Therefore, I need to find the values of x where $x^2 - 36x < 0$ and $|x^2 - 81| \neq 0$.

So, first, let me solve $x^2 - 36x < 0$. Let me factor that:

$x(x - 36) < 0$.

To solve this inequality, I can use a sign chart. The critical points are at x = 0 and x = 36. These divide the number line into three intervals:

1. $x < 0$
2. $0 < x < 36$
3. $x > 36$

Testing each interval:

1. For $x < 0$, let's pick x = -1. Then, (-1)(-1 - 36) = (-1)(-37) = 37 > 0. So, positive.
2. For $0 < x < 36$, let's pick x = 18. Then, 18(18 - 36) = 18(-18) = -324 < 0. So, negative.
3. For $x > 36$, let's pick x = 40. Then, 40(40 - 36) = 40(4) = 160 > 0. So, positive.

Therefore, the inequality $x^2 - 36x < 0$ holds when $0 < x < 36$.

But remember, we also need to exclude the points where the numerator is zero because if the numerator is zero, the entire expression is zero, which is not less than zero. So, the numerator $|x^2 - 81|$ is zero when $x^2 - 81 = 0$, which is when $x^2 = 81$, so $x = 9$ or $x = -9$.

Therefore, in the interval $0 < x < 36$, we need to exclude x = 9 because at x = 9, the numerator is zero, making the entire expression zero, which does not satisfy the strict inequality.

So, putting it all together, the solution set is $0 < x < 36$ excluding x = 9. However, we also need to check if there are any other restrictions. Wait, but the original domain excludes x = 0 and x = 36, which are already excluded in the interval $0 < x < 36$. So, the solution is $0 < x < 36$ with x ≠ 9.

But let me verify this. Let me check if there are any other points where the numerator is zero. The numerator is |x² - 81|, which is zero when x² - 81 = 0, so x = ±9. So, in the interval 0 < x < 36, x = 9 is the only one. x = -9 is not in this interval, so we don't have to worry about that. Therefore, the solution is (0, 9) ∪ (9, 36).

But wait, let me check if there are any other points where the denominator is zero. The denominator is zero at x = 0 and x = 36, which are already excluded. So, the answer is all real numbers between 0 and 36, not including 0, 36, and 9.

But let me test some values to make sure. Let's pick x = 1, which is in (0, 9). Then, numerator is |1 - 81| = | -80 | = 80. Denominator is 1 - 36 = -35. So, 80 / (-35) = -80/35 < 0. So, that works.

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

Now, x = 9. The numerator is zero, so the expression is zero, which is not less than zero. So, x = 9 is excluded. Similarly, x = 0 and x = 36 are excluded because denominator is zero.

What about x = -1? Wait, x = -1 is not in the interval (0, 36), so it's not part of the solution. Let me check. If x = -1, denominator is (-1)^2 - 36*(-1) = 1 + 36 = 37 > 0. So, the fraction is |(-1)^2 - 81| / 37 = |1 - 81| / 37 = 80 / 37 > 0, which is not less than zero. So, that's correct.

But wait, what if x is negative? The denominator is positive for x < 0, so the fraction is non-negative (since numerator is non-negative) divided by positive, so non-negative. Therefore, the inequality is not satisfied for x < 0. Similarly, for x > 36, denominator is positive, so same thing. Therefore, only between 0 and 36, denominator is negative, so the fraction is negative, except when numerator is zero. Therefore, the solution is indeed (0, 9) ∪ (9, 36).

But let me check another point in (0, 9). Let's take x = 5. Numerator: |25 - 81| = 56. Denominator: 25 - 180 = -155. So, 56 / (-155) < 0. Correct.

Another point in (9, 36): x = 18. Numerator: |324 - 81| = 243. Denominator: 324 - 648 = -324. So, 243 / (-324) = -0.75 < 0. Correct.

What about x approaching 0 from the right? Let's say x approaches 0+. Then denominator approaches 0 - 0 = 0? Wait, denominator is x² - 36x. As x approaches 0+, x² approaches 0, and -36x approaches 0 from the negative side. So, denominator approaches 0 from the negative side. The numerator approaches |0 - 81| = 81. Therefore, the expression approaches 81 / 0- which is negative infinity. So, it's negative near x = 0. Similarly, approaching x = 36 from the left, denominator approaches 0 from the negative side. Let me check. Let x approach 36 from the left, so x = 36 - ε, where ε is small positive. Then denominator is (36 - ε)^2 - 36*(36 - ε) = 36² - 72ε + ε² - 36*36 + 36ε = (-72ε + 36ε) + ε² = -36ε + ε² ≈ -36ε < 0. So denominator approaches 0 from the negative side. The numerator is |(36 - ε)^2 - 81|. Let me compute (36 - ε)^2 - 81 = 1296 - 72ε + ε² - 81 = 1215 - 72ε + ε². Since ε is small, this is approximately 1215, which is positive. Therefore, the absolute value is 1215 - 72ε + ε², so the numerator approaches 1215. Therefore, the expression approaches 1215 / 0- which is negative infinity. So, near x = 36, the expression is also negative. Therefore, the entire interval (0, 36) except x = 9 is the solution.

But let me check if there are any other points where the numerator is zero. For example, x = -9. If x = -9, then the numerator is zero, but the denominator is (-9)^2 - 36*(-9) = 81 + 324 = 405 > 0. Therefore, the expression is 0 / 405 = 0, which is not less than zero. Therefore, x = -9 is not part of the solution. So, only x = 9 is excluded in the interval (0, 36).

Therefore, the solution set is all real numbers between 0 and 36, not including 0, 9, and 36. So, in interval notation, that's (0, 9) ∪ (9, 36).

But let me check if there's a possibility that the absolute value could affect the sign in some way. Wait, the absolute value is always non-negative, so the numerator is non-negative. The denominator is negative in (0, 36). Therefore, the entire expression is non-positive (since non-negative divided by negative is non-positive). But the inequality is strict: less than zero. Therefore, we need to exclude the points where the numerator is zero, which is x = ±9. But in the interval (0, 36), only x = 9 is in there. Therefore, the solution is indeed (0, 9) ∪ (9, 36).

Wait, but let me think again. Suppose that the numerator is zero. Then the expression is zero, which is not less than zero. Therefore, we need to exclude x = 9. So, yes, that's correct.

But let me also check if there are any other points where the expression could be undefined. The denominator is zero at x = 0 and x = 36, so those are excluded. So, the answer is (0, 9) ∪ (9, 36).

But let me check if there's a case where the absolute value could make the numerator positive or negative? Wait, no. The absolute value is always non-negative. So, regardless of the value inside, the absolute value is non-negative. Therefore, the numerator is non-negative. Therefore, the sign of the entire expression depends only on the denominator. Therefore, the expression is negative when denominator is negative and numerator is positive, or denominator is negative and numerator is zero? Wait, if numerator is zero, then the expression is zero. Therefore, the inequality is strict, so we need the expression to be negative, which requires denominator negative and numerator positive. Therefore, we need to exclude the points where numerator is zero. Therefore, the solution is denominator negative (which is 0 < x < 36) and numerator not zero (so x ≠ 9). Therefore, the solution is (0, 9) ∪ (9, 36). So, that seems correct.

But let me think again. Suppose that the numerator is zero. Then, the expression is zero. So, the inequality is not satisfied. Therefore, we need to exclude x = 9. Therefore, the answer is (0, 9) ∪ (9, 36). Therefore, I think that is the solution.

But let me check if there's a case where the absolute value could be negative? No, absolute value is always non-negative. Therefore, the numerator is non-negative. Therefore, the fraction is negative only when denominator is negative and numerator is positive. But since numerator is non-negative, the only way for the fraction to be negative is denominator negative and numerator positive. However, if numerator is zero, then the fraction is zero. Therefore, the solution is denominator negative and numerator positive. Therefore, we need to find x where denominator is negative (so 0 < x < 36) and numerator is positive (so |x² - 81| > 0). Which is equivalent to x² - 81 ≠ 0, so x ≠ ±9. Therefore, in the interval 0 < x < 36, x ≠ 9. Therefore, the solution is (0, 9) ∪ (9, 36). Therefore, that's the answer.

But let me check if there's a case where even though the denominator is negative, the numerator is zero. Which is only at x = 9. Therefore, that's the only point to exclude. Therefore, the answer is correct.

Alternatively, maybe I can solve the inequality by considering cases based on the absolute value. Let me try that approach as a check.

The absolute value |x² - 81| can be written as:

|x² - 81| = { x² - 81, if x² - 81 ≥ 0; -(x² - 81), if x² - 81 < 0 }

Which simplifies to:

|x² - 81| = { x² - 81, if |x| ≥ 9; 81 - x², if |x| < 9 }

Therefore, the original inequality is:

Case 1: |x| ≥ 9, so |x² - 81| = x² - 81. Then, the inequality becomes (x² - 81)/(x² - 36x) < 0.

Case 2: |x| < 9, so |x² - 81| = 81 - x². Then, the inequality becomes (81 - x²)/(x² - 36x) < 0.

So, let me analyze both cases.

First, note that the original domain excludes x = 0 and x = 36. Also, in Case 1, |x| ≥ 9, so x ≤ -9 or x ≥ 9. In Case 2, |x| < 9, so -9 < x < 9.

But let me handle each case separately.

Case 1: |x| ≥ 9, so x ≤ -9 or x ≥ 9. The inequality is (x² - 81)/(x² - 36x) < 0.

Let me simplify the expression. Let me factor numerator and denominator.

Numerator: x² - 81 = (x - 9)(x + 9).

Denominator: x² - 36x = x(x - 36).

Therefore, the expression becomes [(x - 9)(x + 9)] / [x(x - 36)] < 0.

So, we need to find where this fraction is negative. Let me note that in Case 1, x ≤ -9 or x ≥ 9. Let me consider these two subcases.

Subcase 1a: x ≤ -9.

In this case, x is negative, and x ≤ -9. Let me check the sign of each factor:

(x - 9): x ≤ -9, so x - 9 ≤ -18 < 0.

(x + 9): x ≤ -9, so x + 9 ≤ 0. But since x ≤ -9, x + 9 ≤ 0. However, when x = -9, x + 9 = 0. So, for x < -9, x + 9 < 0.

x: x ≤ -9, so x < 0.

(x - 36): x ≤ -9, so x - 36 ≤ -45 < 0.

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

Denominator: (negative)(negative) = positive.

Therefore, the entire expression is positive / positive = positive. Therefore, the inequality (positive) < 0 is false. Therefore, no solutions in this subcase.

Subcase 1b: x ≥ 9.

In this case, x ≥ 9. Let me check the sign of each factor:

(x - 9): x ≥ 9, so x - 9 ≥ 0. At x = 9, it's zero.

(x + 9): x ≥ 9, so x + 9 ≥ 18 > 0.

x: x ≥ 9, so x > 0 (since x ≥ 9).

(x - 36): x ≥ 9. So, when x < 36, x - 36 < 0. When x = 36, it's zero. When x > 36, it's positive.

Therefore, the expression [(x - 9)(x + 9)] / [x(x - 36)].

Let me split into intervals:

First, x ≥ 9. The critical points are x = 9, x = 36.

So, intervals:

1. 9 < x < 36

2. x = 9

3. x = 36

4. x > 36

But since we are in Case 1 (x ≥ 9), let's check each interval.

Interval 1: 9 < x < 36.

(x - 9): positive

(x + 9): positive

x: positive

(x - 36): negative

Therefore, numerator: positive * positive = positive

Denominator: positive * negative = negative

Therefore, the expression is positive / negative = negative. Therefore, the inequality holds here. So, 9 < x < 36 is part of the solution.

At x = 9: numerator is zero, so expression is zero. Not less than zero. Excluded.

At x = 36: denominator is zero. Excluded.

Interval 4: x > 36.

(x - 9): positive

(x + 9): positive

x: positive

(x - 36): positive

Therefore, numerator: positive * positive = positive

Denominator: positive * positive = positive

Expression: positive / positive = positive. Not less than zero. Therefore, no solution here.

Therefore, in Case 1, the solution is 9 < x < 36.

Case 2: |x| < 9, so -9 < x < 9. The inequality is (81 - x²)/(x² - 36x) < 0.

Again, let me factor numerator and denominator.

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

Denominator: x² - 36x = x(x - 36).

Therefore, the expression becomes [ - (x - 9)(x + 9) ] / [x(x - 36)] < 0.

Which is equivalent to - (x - 9)(x + 9) / [x(x - 36)] < 0.

Multiply both sides by -1 (remembering to reverse the inequality sign):

(x - 9)(x + 9) / [x(x - 36)] > 0.

So, now we need to solve (x - 9)(x + 9) / [x(x - 36)] > 0.

But in Case 2, we have -9 < x < 9. Let me note that in this interval, x is between -9 and 9, so x ≠ 0 (since denominator is zero at x = 0, which is excluded). Also, x ≠ 9 or -9, but since we are in -9 < x < 9, x ≠ -9 and x ≠ 9 are already excluded.

So, let me analyze the sign of the expression (x - 9)(x + 9) / [x(x - 36)] in the interval -9 < x < 9.

First, let me note that in this interval:

x - 9: since x < 9, x - 9 < 0.

x + 9: since x > -9, x + 9 > 0.

x: in -9 < x < 9, x can be positive or negative. Specifically, x is in (-9, 0) or (0, 9).

x - 36: since x < 9 < 36, x - 36 < 0.

Therefore, let me write the sign of each factor:

(x - 9): negative

(x + 9): positive

x: depends on the subinterval

(x - 36): negative

Therefore, the numerator (x - 9)(x + 9) is negative * positive = negative.

Denominator x(x - 36): x * negative.

So, denominator is x * negative. Let me consider two subcases:

Subcase 2a: -9 < x < 0.

In this case, x is negative. Therefore, denominator: x (negative) * (negative) = positive.

Therefore, the entire expression is numerator (negative) / denominator (positive) = negative. Therefore, the inequality (negative) > 0 is false. Therefore, no solution here.

Subcase 2b: 0 < x < 9.

In this case, x is positive. Therefore, denominator: x (positive) * (negative) = negative.

Therefore, the entire expression is numerator (negative) / denominator (negative) = positive. Therefore, the inequality (positive) > 0 is true. Therefore, this interval is part of the solution.

Therefore, in Case 2, the solution is 0 < x < 9.

Therefore, combining Case 1 and Case 2, the solutions are:

From Case 1: 9 < x < 36

From Case 2: 0 < x < 9

Therefore, the total solution is (0, 9) ∪ (9, 36), which matches my previous conclusion.

Therefore, the answer is all real numbers between 0 and 36, excluding 0, 9, and 36. Therefore, in interval notation, that is (0, 9) ∪ (9, 36).

But let me check if there are any other points where the expression could be negative. For example, in Case 2, when we had 0 < x < 9, the expression was positive? Wait, no. Wait, in Case 2, after multiplying by -1, we had the inequality (x - 9)(x + 9)/[x(x - 36)] > 0. Then, in Subcase 2b, 0 < x < 9, the expression was positive, so the original inequality (before multiplying by -1) was negative. Wait, let me retrace.

Original Case 2 inequality: (81 - x²)/(x² - 36x) < 0.

Which became - (x - 9)(x + 9)/[x(x - 36)] < 0.

Then, multiplying both sides by -1 reverses the inequality:

(x - 9)(x + 9)/[x(x - 36)] > 0.

So, solving this inequality gives us the solution for Case 2. In Subcase 2b, 0 < x < 9, we found that the expression (x - 9)(x + 9)/[x(x - 36)] is positive, so the original inequality holds. Therefore, in Case 2, 0 < x < 9 is part of the solution. Therefore, combining with Case 1, the total solution is (0, 9) ∪ (9, 36). Therefore, that's correct.

Therefore, after checking both cases, I confirm that the solution is (0, 9) ∪ (9, 36).

But let me check if there's any mistake in the sign analysis. For example, in Case 2, when I had 0 < x < 9, the expression (x - 9)(x + 9)/[x(x - 36)]:

(x - 9) is negative, (x + 9) is positive, x is positive, (x - 36) is negative. Therefore, numerator: negative * positive = negative. Denominator: positive * negative = negative. Therefore, negative / negative = positive. Therefore, the inequality (positive) > 0 is true. Therefore, original inequality holds. Therefore, correct.

Therefore, the answer is indeed (0, 9) ∪ (9, 36).

Therefore, I think this is the correct solution.

**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 proceed step by step, analyzing the behavior of the numerator and denominator.

---

### **Step 1: Domain Considerations**

The expression is undefined when the denominator is zero.
$$
x^2 - 36x = x(x - 36) = 0 \Rightarrow x = 0 \text{ or } x = 36.
$$
So, the domain of the expression is all real numbers **except** $ x = 0 $ and $ x = 36 $.

---

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

The numerator is an absolute value:
$$
|x^2 - 81| \geq 0 \quad \text{for all } x.
$$
So, the numerator is **non-negative**.
The inequality is strict:
$$
\frac{|x^2 - 81|}{x^2 - 36x} < 0.
$$
This means the **denominator must be negative**, and the **numerator must not be zero** (since a zero numerator would make the expression zero, not negative).

So, we need to find the values of $ x $ such that:
- $ x^2 - 36x < 0 $, and
- $ |x^2 - 81| \neq 0 $.

---

### **Step 3: Solve the Inequality $ x^2 - 36x < 0 $**

Factor the denominator:
$$
x^2 - 36x = x(x - 36).
$$
This is a quadratic that opens upwards, with roots at $ x = 0 $ and $ x = 36 $.
Using a sign chart or test points, we find:
- $ x < 0 $: positive
- $ 0 < x < 36 $: negative
- $ x > 36 $: positive

So, the inequality $ x^2 - 36x < 0 $ holds for:
$$
0 < x < 36.
$$

---

### **Step 4: Exclude Points Where the Numerator is Zero**

The numerator is zero when:
$$
|x^2 - 81| = 0 \Rightarrow x^2 - 81 = 0 \Rightarrow x = \pm 9.
$$
So, we must **exclude** $ x = 9 $ from the interval $ (0, 36) $, because at $ x = 9 $, the expression becomes zero, which does **not** satisfy the strict inequality.

---

### **Step 5: Final Solution**

Combining the above:
- The expression is negative when $ 0 < x < 36 $,
- But we must exclude $ x = 9 $ (where the numerator is zero),
- And we must also exclude $ x = 0 $ and $ x = 36 $ (where the expression is undefined).

Thus, the solution set is:
$$
(0, 9) \cup (9, 36).
$$

---

### **Final Answer**

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

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.

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 side. Let me do that:
$$ x^2 - 5x - 4 - 10 \leq 0 $$
Simplifying the constants:
$$ x^2 - 5x - 14 \leq 0 $$

Okay, so now the inequality is $ x^2 - 5x - 14 \leq 0 $. Next step is to find the roots of the quadratic equation $ x^2 - 5x - 14 = 0 $. Once I have the roots, I can determine the intervals where the quadratic is less than 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 $.

Plugging in the values:
$$ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} $$
Simplify step by step:
First, calculate the discriminant:
$$ (-5)^2 - 4(1)(-14) = 25 + 56 = 81 $$

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

So continuing with the quadratic formula:
$$ x = \frac{5 \pm \sqrt{81}}{2} $$
$$ \sqrt{81} = 9 $$
Therefore:
$$ x = \frac{5 \pm 9}{2} $$

Calculating both roots:
First root with the plus sign:
$$ x = \frac{5 + 9}{2} = \frac{14}{2} = 7 $$
Second root with the minus sign:
$$ x = \frac{5 - 9}{2} = \frac{-4}{2} = -2 $$

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

Now, since the quadratic coefficient $ a = 1 $ is positive, the parabola opens upwards. This means the quadratic will be below or equal to zero between its two roots.

Wait, let me confirm that. If the parabola opens upwards, it will be below the x-axis (negative) between the roots and above the x-axis outside the roots. Therefore, the inequality $ x^2 - 5x - 14 \leq 0 $ is satisfied for all x between -2 and 7, including the endpoints because of the less than or equal to sign.

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

But let me double-check my steps to make sure I didn't make a mistake.

Original inequality: $ x^2 - 5x - 4 \leq 10 $. Subtract 10: $ x^2 - 5x - 14 \leq 0 $. Correct.

Quadratic formula: coefficients a=1, b=-5, c=-14. Discriminant: b² - 4ac = 25 - 4(1)(-14) = 25 + 56 = 81. Correct. Square root of 81 is 9. Then x = [5 ± 9]/2. So 5 + 9 is 14 over 2 is 7, 5 - 9 is -4 over 2 is -2. Correct.

Parabola opens upwards, so the inequality is satisfied between the roots. So the interval is from -2 to 7, including both endpoints.

Wait, but let me test a value in each interval to make sure.

First, let's test x = -3, which is less than -2. Plug into the original inequality:

Original inequality was $ x^2 - 5x - 4 \leq 10 $. Let me compute left side for x = -3:

(-3)^2 -5*(-3) -4 = 9 +15 -4 = 20. Is 20 ≤ 10? No. So that's outside the solution set.

Now test x = 0, which is between -2 and 7. Compute left side:

0^2 -5*0 -4 = 0 - 0 -4 = -4. Is -4 ≤ 10? Yes. So that's good.

Now test x = 8, which is greater than 7. Compute left side:

8^2 -5*8 -4 = 64 - 40 -4 = 20. Is 20 ≤ 10? No.

So that seems to confirm that between -2 and 7, the inequality holds.

But just to be thorough, let me check the endpoints.

At x = -2: (-2)^2 -5*(-2) -4 = 4 +10 -4 = 10. Which is equal to 10, so satisfies the inequality.

At x = 7: 7^2 -5*7 -4 = 49 -35 -4 = 10. Also equal to 10, so satisfies the inequality.

Therefore, the solution is indeed all real numbers from -2 to 7, inclusive.

So in interval notation, that is [-2, 7].

I think that's all. I don't see any mistakes in my reasoning. Let me just recap:

1. Move all terms to left side: correct.
2. Find roots using quadratic formula: correct.
3. Determine direction of parabola: correct.
4. Test intervals: correct.
5. Check endpoints: correct.

Therefore, the answer is [-2, 7].

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

To solve the inequality:

$$
x^2 - 5x - 4 \leq 10
$$

we begin by moving all terms to one side of the inequality to set it equal to zero:

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

Simplifying the constants:

$$
x^2 - 5x - 14 \leq 0
$$

---

### Step 1: Find the Roots

We solve the corresponding quadratic equation:

$$
x^2 - 5x - 14 = 0
$$

Using the quadratic formula:

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

$$
x = \frac{5 \pm 9}{2}
$$

This gives two roots:

$$
x = \frac{5 + 9}{2} = 7 \quad \text{and} \quad x = \frac{5 - 9}{2} = -2
$$

---

### Step 2: Analyze the Inequality

The inequality is:

$$
x^2 - 5x - 14 \leq 0
$$

This is a quadratic inequality. Since the leading coefficient is positive (1), the parabola opens **upwards**. Therefore, the expression is **less than or equal to zero** between the two roots.

So the solution to the inequality is the interval:

$$
[-2, 7]
$$

---

### Step 3: Verify the Solution

To confirm, we can test values in different intervals:

- For $ x = -3 $ (left of -2):
  $ (-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 $, which is **not** ≤ 0.

- For $ x = 0 $ (between -2 and 7):
  $ 0^2 - 5(0) - 14 = -14 $, which is ≤ 0.

- For $ x = 8 $ (right of 7):
  $ 8^2 - 5(8) - 14 = 64 - 40 - 14 = 10 $, which is **not** ≤ 0.

Also, the endpoints $ x = -2 $ and $ x = 7 $ yield:

- $ (-2)^2 - 5(-2) - 14 = 4 + 10 - 14 = 0 $
- $ 7^2 - 5(7) - 14 = 49 - 35 - 14 = 0 $

Both satisfy the inequality.

---

### 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.

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 side. Let me do that:
$$ x^2 - 5x - 4 - 10 \leq 0 $$
Simplifying the constants:
$$ x^2 - 5x - 14 \leq 0 $$

Okay, so now the inequality is $ x^2 - 5x - 14 \leq 0 $. Next step is to find the roots of the quadratic equation $ x^2 - 5x - 14 = 0 $. Once I have the roots, I can determine the intervals where the quadratic is less than 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 $.

Plugging in the values:
$$ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} $$
Simplify step by step:
First, calculate the discriminant:
$$ (-5)^2 - 4(1)(-14) = 25 + 56 = 81 $$

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

So continuing with the quadratic formula:
$$ x = \frac{5 \pm \sqrt{81}}{2} $$
$$ \sqrt{81} = 9 $$
Therefore:
$$ x = \frac{5 \pm 9}{2} $$

Calculating both roots:
First root with the plus sign:
$$ x = \frac{5 + 9}{2} = \frac{14}{2} = 7 $$
Second root with the minus sign:
$$ x = \frac{5 - 9}{2} = \frac{-4}{2} = -2 $$

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

Now, since the quadratic coefficient $ a = 1 $ is positive, the parabola opens upwards. This means the quadratic will be below or equal to zero between its two roots.

Wait, let me confirm that. If the parabola opens upwards, it will be below the x-axis (negative) between the roots and above the x-axis outside the roots. Therefore, the inequality $ x^2 - 5x - 14 \leq 0 $ is satisfied for all x between -2 and 7, including the endpoints because of the less than or equal to sign.

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

But let me double-check my steps to make sure I didn't make a mistake.

Original inequality: $ x^2 - 5x - 4 \leq 10 $. Subtract 10: $ x^2 - 5x - 14 \leq 0 $. Correct.

Quadratic formula: coefficients a=1, b=-5, c=-14. Discriminant: b² - 4ac = 25 - 4(1)(-14) = 25 + 56 = 81. Correct. Square root of 81 is 9. Then x = [5 ± 9]/2. So 5 + 9 is 14 over 2 is 7, 5 - 9 is -4 over 2 is -2. Correct.

Parabola opens upwards, so the inequality is satisfied between the roots. So the interval is from -2 to 7, including both endpoints.

Wait, but let me test a value in each interval to make sure.

First, let's test x = -3, which is less than -2. Plug into the original inequality:

Original inequality was $ x^2 - 5x - 4 \leq 10 $. Let me compute left side for x = -3:

(-3)^2 -5*(-3) -4 = 9 +15 -4 = 20. Is 20 ≤ 10? No. So that's outside the solution set.

Now test x = 0, which is between -2 and 7. Compute left side:

0^2 -5*0 -4 = 0 - 0 -4 = -4. Is -4 ≤ 10? Yes. So that's good.

Now test x = 8, which is greater than 7. Compute left side:

8^2 -5*8 -4 = 64 - 40 -4 = 20. Is 20 ≤ 10? No.

So that seems to confirm that between -2 and 7, the inequality holds.

But just to be thorough, let me check the endpoints.

At x = -2: (-2)^2 -5*(-2) -4 = 4 +10 -4 = 10. Which is equal to 10, so satisfies the inequality.

At x = 7: 7^2 -5*7 -4 = 49 -35 -4 = 10. Also equal to 10, so satisfies the inequality.

Therefore, the solution is indeed all real numbers from -2 to 7, inclusive.

So in interval notation, that is [-2, 7].

I think that's all. I don't see any mistakes in my reasoning. Let me just recap:

1. Move all terms to left side: correct.
2. Find roots using quadratic formula: correct.
3. Determine direction of parabola: correct.
4. Test intervals: correct.
5. Check endpoints: correct.

Therefore, the answer is [-2, 7].

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

To solve the inequality:

$$
x^2 - 5x - 4 \leq 10
$$

we begin by moving all terms to one side of the inequality to set it equal to zero:

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

Simplifying the constants:

$$
x^2 - 5x - 14 \leq 0
$$

---

### Step 1: Find the Roots

We solve the corresponding quadratic equation:

$$
x^2 - 5x - 14 = 0
$$

Using the quadratic formula:

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

$$
x = \frac{5 \pm 9}{2}
$$

This gives two roots:

$$
x = \frac{5 + 9}{2} = 7 \quad \text{and} \quad x = \frac{5 - 9}{2} = -2
$$

---

### Step 2: Analyze the Inequality

The inequality is:

$$
x^2 - 5x - 14 \leq 0
$$

This is a quadratic inequality. Since the leading coefficient is positive (1), the parabola opens **upwards**. Therefore, the expression is **less than or equal to zero** between the two roots.

So the solution to the inequality is the interval:

$$
[-2, 7]
$$

---

### Step 3: Verify the Solution

To confirm, we can test values in different intervals:

- For $ x = -3 $ (left of -2):
  $ (-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 $, which is **not** ≤ 0.

- For $ x = 0 $ (between -2 and 7):
  $ 0^2 - 5(0) - 14 = -14 $, which is ≤ 0.

- For $ x = 8 $ (right of 7):
  $ 8^2 - 5(8) - 14 = 64 - 40 - 14 = 10 $, which is **not** ≤ 0.

Also, the endpoints $ x = -2 $ and $ x = 7 $ yield:

- $ (-2)^2 - 5(-2) - 14 = 4 + 10 - 14 = 0 $
- $ 7^2 - 5(7) - 14 = 49 - 35 - 14 = 0 $

Both satisfy the inequality.

---

### 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, so Navin, Luka, and Ian are talking about their part-time jobs, and we need to figure out who earns the most after working an eight-hour day. The problem gives their hourly wages in their local currencies: Navin makes 160 rupee per hour, Luka makes 25 kuna per hour, and Ian makes 34 pula per hour. Then it gives the exchange rates for US dollars to each of their currencies. The question is, who earns the most after working 8 hours?

Alright, so first, I need to convert each of their hourly wages into US dollars, then multiply by 8 hours to get their daily earnings in USD. Then compare those amounts to see who has the highest. That seems like the plan. Let me write that down step by step.

First, let me note down the given information:

- Navin's wage: 160 rupee per hour
- Luka's wage: 25 kuna per hour
- Ian's wage: 34 pula per hour

Exchange rates:

- 1 USD = 32.35 Mauritius rupee
- 1 USD = 5.18 Croatian kuna
- 1 USD = 6.95 Botswana pula

So, to convert their wages to USD, I need to divide their hourly wage by the exchange rate. Because if 1 USD is equal to X local currency, then 1 local currency is equal to 1/X USD. For example, if 1 USD = 32.35 rupee, then 1 rupee is 1/32.35 USD. Therefore, Navin's hourly wage in USD would be 160 divided by 32.35. Similarly for the others.

Once I have their hourly wages in USD, multiply by 8 hours to get the total earnings for the day. Then compare the three totals.

Let me compute each one step by step.

Starting with Navin:

Navin's hourly wage in rupee: 160 rupee/hour

Exchange rate: 1 USD = 32.35 rupee

So, to convert to USD: 160 / 32.35

Let me calculate that. Let me get a calculator or do it manually. Let me approximate.

First, 32.35 times 5 is 161.75. Wait, 32.35 * 5 = 161.75. But Navin's wage is 160, which is slightly less than 161.75. So 160 / 32.35 is approximately 4.947... USD per hour? Wait, because 32.35 * 4.947 ≈ 160?

Wait, let me check:

Let me compute 160 divided by 32.35.

Let me write it as 160 / 32.35.

Let me divide numerator and denominator by 5 to simplify? Let me see:

32.35 divided by 5 is 6.47, and 160 divided by 5 is 32. So 32 / 6.47 ≈ ?

Well, 6.47 * 5 = 32.35, which is slightly more than 32. So 32 / 6.47 ≈ 4.947. So approximately 4.947 USD per hour. Let me check with actual division.

Alternatively, maybe I can use decimal division.

Alternatively, since 32.35 * 4.9 = ?

Let me compute 32.35 * 4.9:

First, 32.35 * 4 = 129.4

32.35 * 0.9 = 29.115

Adding together: 129.4 + 29.115 = 158.515

So 32.35 * 4.9 = 158.515

But Navin's wage is 160, which is 160 - 158.515 = 1.485 more.

So how much more is that? Since 32.35 * x = 1.485, so x ≈ 1.485 / 32.35 ≈ 0.0459

Therefore, total is approximately 4.9 + 0.0459 ≈ 4.9459 USD per hour. So approximately 4.946 USD per hour.

So Navin's hourly wage in USD is approximately 4.946 USD.

Therefore, for 8 hours, Navin earns 4.946 * 8.

Let me compute that. 4.946 * 8:

Well, 4 * 8 = 32, 0.946 * 8 = 7.568, so total is 32 + 7.568 = 39.568 USD. So approximately 39.57 USD per day.

Okay, that's Navin.

Now moving on to Luka.

Luka's wage: 25 kuna per hour.

Exchange rate: 1 USD = 5.18 kuna.

Therefore, to convert to USD: 25 / 5.18.

Let me compute that. Let me see.

First, 5.18 * 4 = 20.72

Subtract that from 25: 25 - 20.72 = 4.28

So that's 4 + (4.28 / 5.18). Let me compute 4.28 / 5.18.

Well, 5.18 * 0.8 = 4.144

So 4.28 - 4.144 = 0.136

So that's 0.8 + (0.136 / 5.18). 0.136 / 5.18 ≈ 0.0263

So total is approximately 4.8 + 0.0263 ≈ 4.8263 USD per hour.

Wait, let me check again:

Wait, original value is 25 / 5.18. Let me do this division properly.

Alternatively, approximate decimal.

Let me note that 5.18 * 4.8 = ?

Well, 5 * 4.8 = 24, 0.18 * 4.8 = 0.864, so total is 24 + 0.864 = 24.864

Which is less than 25. The difference is 25 - 24.864 = 0.136.

So, as before, so 4.8 + (0.136 / 5.18). Which is approximately 4.8 + 0.0263 ≈ 4.8263 USD per hour.

So approximately 4.826 USD per hour.

Therefore, for 8 hours, Luka earns 4.826 * 8.

Let me compute that. 4 * 8 = 32, 0.826 * 8 = 6.608, so total is 32 + 6.608 = 38.608 USD. Approximately 38.61 USD per day.

So Luka earns about 38.61 USD per day.

Now Ian.

Ian's wage: 34 pula per hour.

Exchange rate: 1 USD = 6.95 pula.

Therefore, converting to USD: 34 / 6.95.

Let me compute that. Let me see.

First, 6.95 * 4 = 27.8

Subtract that from 34: 34 - 27.8 = 6.2

So that's 4 + (6.2 / 6.95). Let me compute 6.2 / 6.95.

Well, 6.95 * 0.89 ≈ ?

Wait, 6.95 * 0.9 = 6.255. Which is slightly more than 6.2. So 0.9 - (6.255 - 6.2)/6.95.

Wait, 6.255 - 6.2 = 0.055. So 0.055 / 6.95 ≈ 0.0079. Therefore, approximately 0.9 - 0.0079 ≈ 0.8921.

Therefore, total is approximately 4 + 0.8921 ≈ 4.8921 USD per hour.

Alternatively, let me check with another method.

Let me compute 34 divided by 6.95.

Let me write this as 34 / 6.95. Multiply numerator and denominator by 100 to eliminate decimals: 3400 / 695.

Simplify this fraction. Let me divide numerator and denominator by 5: 3400 / 5 = 680, 695 / 5 = 139. So 680 / 139.

Now, let me divide 680 by 139.

139 * 4 = 556

Subtract from 680: 680 - 556 = 124

Bring down a zero (making it 1240). 139 * 8 = 1112

Subtract: 1240 - 1112 = 128

Bring down another zero: 1280. 139 * 9 = 1251

Subtract: 1280 - 1251 = 29

So so far, we have 4.89... So approximately 4.89 USD per hour.

So approximately 4.89 USD per hour.

Therefore, for 8 hours, Ian earns 4.89 * 8.

Let me compute that. 4 * 8 = 32, 0.89 * 8 = 7.12, so total is 32 + 7.12 = 39.12 USD per day.

So Ian earns approximately 39.12 USD per day.

So summarizing:

Navin: ~39.57 USD

Luka: ~38.61 USD

Ian: ~39.12 USD

Therefore, Navin earns the most, followed by Ian, then Luka.

Wait, but let me check my calculations again because approximations can sometimes lead to errors. Let me verify each calculation more precisely.

Starting with Navin:

160 rupee per hour. 1 USD = 32.35 rupee. So 160 / 32.35.

Let me compute this division more accurately.

Let me use decimal division.

Let me write 160 ÷ 32.35.

First, note that 32.35 × 4 = 129.4

Subtract that from 160: 160 - 129.4 = 30.6

Bring down a decimal: So we have 30.6 divided by 32.35. Let me add decimal places.

So 30.6 / 32.35 = 0.945... Wait, but actually, since we have 4.??? So total is 4 + (30.6 / 32.35). Let me compute 30.6 / 32.35.

Multiply numerator and denominator by 100: 3060 / 3235.

Divide numerator and denominator by 5: 612 / 647 ≈ ?

Well, 647 × 0.945 ≈ ?

Wait, maybe another approach. Let me approximate 30.6 / 32.35 ≈ 0.945. Wait, because 32.35 × 0.945 ≈ 32.35 × 0.9 + 32.35 × 0.045 = 29.115 + 1.45575 ≈ 30.57075. Which is very close to 30.6. Therefore, 30.6 / 32.35 ≈ 0.945. Therefore, total is 4.945 USD per hour. So 4.945 * 8 = ?

4.945 * 8:

Well, 4 * 8 = 32, 0.945 * 8 = 7.56, so total is 32 + 7.56 = 39.56 USD. So Navin's total is approximately 39.56 USD.

Earlier I had 39.57, which is consistent.

Now Luka: 25 kuna per hour. 1 USD = 5.18 kuna. So 25 / 5.18.

Let me compute this division more accurately.

Let me do 25 ÷ 5.18.

First, 5.18 × 4 = 20.72

Subtract from 25: 25 - 20.72 = 4.28

Bring decimal places. So now, 4.28 / 5.18.

Multiply numerator and denominator by 100: 428 / 518.

Divide numerator and denominator by 2: 214 / 259 ≈ ?

Let me compute 214 ÷ 259.

Well, 259 × 0.8 = 207.2

Subtract from 214: 214 - 207.2 = 6.8

So that's 0.8 + (6.8 / 259). 6.8 / 259 ≈ 0.02625.

Therefore, total is approximately 0.82625. Therefore, total value is 4 + 0.82625 ≈ 4.82625 USD per hour.

Therefore, 4.82625 * 8 = ?

4 * 8 = 32, 0.82625 * 8 = 6.61, so total is 32 + 6.61 = 38.61 USD. Which matches my previous calculation.

Now Ian: 34 pula per hour. 1 USD = 6.95 pula. So 34 / 6.95.

Let me compute this division more accurately.

34 ÷ 6.95.

First, 6.95 × 4 = 27.8

Subtract from 34: 34 - 27.8 = 6.2

Bring decimal places. So 6.2 / 6.95.

Multiply numerator and denominator by 100: 620 / 695.

Divide numerator and denominator by 5: 124 / 139 ≈ ?

Let me compute 124 ÷ 139.

Well, 139 × 0.89 = ?

139 × 0.8 = 111.2

139 × 0.09 = 12.51

Total: 111.2 + 12.51 = 123.71

Which is very close to 124. The difference is 124 - 123.71 = 0.29.

So 0.89 + (0.29 / 139) ≈ 0.89 + 0.00208 ≈ 0.89208.

Therefore, total value is 4 + 0.89208 ≈ 4.89208 USD per hour.

Therefore, 4.89208 * 8 = ?

4 * 8 = 32, 0.89208 * 8 = ?

Well, 0.8 * 8 = 6.4, 0.09208 * 8 ≈ 0.73664. So total is 6.4 + 0.73664 ≈ 7.13664. Therefore, total earnings: 32 + 7.13664 ≈ 39.13664 USD. Approximately 39.14 USD.

So Ian earns approximately 39.14 USD per day.

So comparing:

Navin: ~39.56 USD

Ian: ~39.14 USD

Luka: ~38.61 USD

Therefore, Navin earns the most.

But let me check once again if there's any miscalculations. Let me verify with exact decimal computations or perhaps use fractions.

Alternatively, maybe I can use calculator-like computations.

Alternatively, let me check Navin's rate again.

Navin: 160 / 32.35.

Let me compute 160 / 32.35.

Let me note that 32.35 × 4.94 = ?

Well, 32.35 × 4 = 129.4

32.35 × 0.9 = 29.115

32.35 × 0.04 = 1.294

So total for 4.94 is 129.4 + 29.115 + 1.294 = 129.4 + 30.409 = 159.809

Which is very close to 160. The difference is 160 - 159.809 = 0.191.

So, 0.191 / 32.35 ≈ 0.0059. Therefore, total is 4.94 + 0.0059 ≈ 4.9459 USD per hour. So 4.9459 * 8 = 39.5672 USD. So approximately 39.57 USD. Correct.

For Ian: 34 / 6.95.

Let me compute 34 / 6.95.

Let me note that 6.95 × 4.89 = ?

Well, 6.95 × 4 = 27.8

6.95 × 0.8 = 5.56

6.95 × 0.09 = 0.6255

Adding those: 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is very close to 34. The difference is 34 - 33.9855 = 0.0145.

So, 0.0145 / 6.95 ≈ 0.002086. Therefore, total is 4.89 + 0.002086 ≈ 4.892086 USD per hour. Multiply by 8: 4.892086 * 8 ≈ 39.1367 USD. So approximately 39.14 USD. Correct.

So Navin is at ~39.57, Ian ~39.14, Luka ~38.61. Therefore, Navin earns the most.

But let me check if there's any chance that my approximated decimal calculations might have errors. For example, maybe I should carry out more precise calculations or use fractions?

Alternatively, perhaps I can use exact fractions with the given exchange rates.

But given that the exchange rates are given to two decimal places, maybe the question expects us to use those exact rates and compute accordingly. Let me check if there's any possibility that rounding during the calculation might affect the result.

Alternatively, maybe I can compute each value with more precision.

Let me try Navin again:

Navin's hourly wage in USD: 160 / 32.35

Let me compute this division with more precision.

Let me write 160 ÷ 32.35.

Let me convert this to 16000 ÷ 3235 (multiplying numerator and denominator by 100 to eliminate decimals).

Now, let me perform the division 16000 ÷ 3235.

First, 3235 × 4 = 12940

Subtract from 16000: 16000 - 12940 = 3060

Bring down a zero (making it 30600). Now divide 30600 by 3235.

3235 × 9 = 29115

Subtract from 30600: 30600 - 29115 = 1485

Bring down a zero (making it 14850). Divide by 3235.

3235 × 4 = 12940

Subtract: 14850 - 12940 = 1910

Bring down a zero (making it 19100). Divide by 3235.

3235 × 5 = 16175

Subtract: 19100 - 16175 = 2925

Bring down a zero (making it 29250). Divide by 3235.

3235 × 9 = 29115

Subtract: 29250 - 29115 = 135

So so far, we have 4.9459... So the value is approximately 4.9459 USD per hour. Therefore, 4.9459 * 8 = ?

Let me compute 4.9459 * 8:

First, 4 * 8 = 32

0.9459 * 8:

0.9 * 8 = 7.2

0.0459 * 8 = 0.3672

Total: 7.2 + 0.3672 = 7.5672

Therefore, total earnings: 32 + 7.5672 = 39.5672 USD. So approximately 39.57 USD. Correct.

For Ian: 34 / 6.95.

Let me compute 34 ÷ 6.95.

Convert to 3400 ÷ 695.

Divide numerator and denominator by 5: 680 ÷ 139.

Let me perform this division.

139 × 4 = 556

Subtract from 680: 680 - 556 = 124

Bring down a zero: 1240

139 × 8 = 1112

Subtract: 1240 - 1112 = 128

Bring down a zero: 1280

139 × 9 = 1251

Subtract: 1280 - 1251 = 29

Bring down a zero: 290

139 × 2 = 278

Subtract: 290 - 278 = 12

Bring down a zero: 120

139 × 0 = 0

So so far, we have 4.892... So the value is approximately 4.892 USD per hour. Multiply by 8:

4.892 * 8 = ?

Well, 4 * 8 = 32, 0.892 * 8 = 7.136, so total is 32 + 7.136 = 39.136 USD. So approximately 39.14 USD.

Therefore, Navin is at ~39.57, Ian ~39.14, Luka ~38.61. So Navin is the highest.

But let me check once again for Luka, just to be thorough.

Luka: 25 / 5.18.

Convert to 2500 / 518.

Divide numerator and denominator by 2: 1250 / 259.

Let me perform this division.

259 × 4 = 1036

Subtract from 1250: 1250 - 1036 = 214

Bring down a zero: 2140

259 × 8 = 2072

Subtract: 2140 - 2072 = 68

Bring down a zero: 680

259 × 2 = 518

Subtract: 680 - 518 = 162

Bring down a zero: 1620

259 × 6 = 1554

Subtract: 1620 - 1554 = 66

Bring down a zero: 660

259 × 2 = 518

Subtract: 660 - 518 = 142

So so far, we have 4.826... So the value is approximately 4.826 USD per hour. Multiply by 8:

4.826 * 8 = ?

4 * 8 = 32, 0.826 * 8 = 6.608, total 38.608 USD. So ~38.61 USD. Correct.

Therefore, the calculations are consistent.

Therefore, Navin earns the most, followed by Ian, then Luka.

But let me check if there's any chance that the question expects us to consider something else, like maybe the exchange rates are given as USD to local currency, so maybe we need to check if the question is about converting their earnings to USD or comparing in their own currencies? Wait, no, the question says "who would earn the most after working an eight-hour day?" and since they are discussing their part-time jobs, but the question is likely asking in USD terms because the exchange rates are given. Otherwise, if we compare in their own currencies, we can't compare because they are different currencies. So the question must be asking for the earnings converted to USD.

Alternatively, maybe the question is asking in their own currencies? But that wouldn't make sense because the currencies are different. So the answer must be in USD.

Alternatively, maybe the question is asking who earns the most in their own currency? But that would be impossible to compare unless we have exchange rates. But since exchange rates are given, it's more likely that we need to convert to USD.

Alternatively, maybe the question is asking for the total earnings in their own currency? But then, since they are different currencies, we can't compare. Therefore, the only logical way is to convert to USD.

Therefore, the answer is Navin.

But let me check once again if I did the calculations correctly. Maybe I made a mistake in the exchange rates?

Wait, the exchange rates are:

1 USD = 32.35 Mauritius rupee

So to get rupee to USD, divide by 32.35. Correct.

Similarly for others. Correct.

So, yes, the calculations are correct.

Therefore, Navin earns the most.

But let me check once again with exact decimal values.

Alternatively, maybe I can use calculator-like computations for each.

For Navin:

160 / 32.35 = ?

Let me approximate 32.35 × 4.94 = ?

As before, 32.35 × 4.94 ≈ 159.8, which is very close to 160. So 4.94 USD per hour. So 4.94 × 8 = 39.52 USD. But earlier precise calculation gave 39.5672. So depending on rounding, maybe 39.57.

For Ian:

34 / 6.95 ≈ 4.892 USD per hour. 4.892 × 8 = 39.136 USD.

So Navin is higher.

Therefore, the conclusion is Navin earns the most.

But let me check if there's any chance that the question is in another way. For example, maybe the exchange rates are given as USD to local currency, so to find the earnings in USD, we need to multiply their local currency earnings by the exchange rate? Wait, no. Wait, if 1 USD = X local currency, then 1 local currency = 1/X USD. Therefore, to convert local currency to USD, you divide by X. So for example, if you have Y local currency, then in USD it's Y / X.

Alternatively, if the exchange rate was given as local currency per USD, then you would multiply. But the question says "one US dollar is equivalent to 32.35 Mauritius rupee", meaning 1 USD = 32.35 rupee. Therefore, to get USD from rupee, divide by 32.35. Correct.

Therefore, calculations are correct.

Therefore, the answer is Navin.

But let me check once again with exact numbers.

Navin: 160 / 32.35 = ?

Let me compute 160 / 32.35:

Let me use cross multiplication.

Let me consider that 32.35 × 4.94 = ?

As above, 32.35 × 4.94 = 32.35 × (4 + 0.9 + 0.04) = 32.35×4 + 32.35×0.9 + 32.35×0.04

= 129.4 + 29.115 + 1.294 = 129.4 + 29.115 = 158.515 + 1.294 = 159.809

Which is 159.809, which is 0.191 less than 160. So, to get the exact value, we can write:

Let me denote x = 160 / 32.35

We have 32.35 × x = 160

We know that 32.35 × 4.94 = 159.809

So, 32.35 × (4.94 + Δ) = 160

So, 32.35 × Δ = 160 - 159.809 = 0.191

Therefore, Δ = 0.191 / 32.35 ≈ 0.0059

Therefore, x ≈ 4.94 + 0.0059 ≈ 4.9459 USD per hour.

Therefore, 4.9459 × 8 = 39.5672 USD.

Similarly, for Ian:

34 / 6.95 = ?

Let me compute 6.95 × 4.89 = ?

As above, 6.95 × 4.89 = 6.95 × (4 + 0.8 + 0.09) = 6.95×4 + 6.95×0.8 + 6.95×0.09

= 27.8 + 5.56 + 0.6255 = 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is 33.9855, which is 0.0145 less than 34. Therefore, Δ = 0.0145 / 6.95 ≈ 0.002086

Therefore, x ≈ 4.89 + 0.002086 ≈ 4.892086 USD per hour.

Multiply by 8: 4.892086 × 8 = 39.1367 USD.

So, yes, Navin is higher.

Therefore, the answer is Navin.

But let me check once again if there's any possible mistake in the exchange rates. For example, if I confused the exchange rates?

Wait, the problem says:

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

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

Similarly, 1 USD = 5.18 kuna, so kuna to USD is divide by 5.18. Correct.

Same for pula. Correct.

Therefore, calculations are correct.

Therefore, Navin earns the most.

But just to make sure, let me check with another approach. Let me compute all three in USD:

Navin: 160 / 32.35 ≈ 4.9459 USD/hour

Luka: 25 / 5.18 ≈ 4.8263 USD/hour

Ian: 34 / 6.95 ≈ 4.8921 USD/hour

Then multiply by 8:

Navin: ~39.57 USD

Luka: ~38.61 USD

Ian: ~39.14 USD

Yes, so Navin is the highest.

Therefore, the answer is Navin.

**Final Answer**
\boxed{Navin}

To determine who earns the most after working an eight-hour day, we need to convert each person's hourly wage from their local currency to US dollars (USD) using the given exchange rates. Then, we multiply each hourly wage in USD by 8 to find the total daily earnings in USD. Finally, we compare the results.

---

### Step 1: Convert Hourly Wages to USD

#### Navin (Mauritius Rupee)
- Hourly wage: 160 rupee
- Exchange rate: 1 USD = 32.35 rupee
- USD per hour:
  $$
  \frac{160}{32.35} \approx 4.9459 \text{ USD}
  $$

#### Luka (Croatian Kuna)
- Hourly wage: 25 kuna
- Exchange rate: 1 USD = 5.18 kuna
- USD per hour:
  $$
  \frac{25}{5.18} \approx 4.8263 \text{ USD}
  $$

#### Ian (Botswana Pula)
- Hourly wage: 34 pula
- Exchange rate: 1 USD = 6.95 pula
- USD per hour:
  $$
  \frac{34}{6.95} \approx 4.8921 \text{ USD}
  $$

---

### Step 2: Calculate Daily Earnings (8 hours)

#### Navin
$$
4.9459 \times 8 \approx 39.57 \text{ USD}
$$

#### Luka
$$
4.8263 \times 8 \approx 38.61 \text{ USD}
$$

#### Ian
$$
4.8921 \times 8 \approx 39.14 \text{ USD}
$$

---

### Step 3: Compare Earnings

- Navin: ≈ **39.57 USD**
- Ian: ≈ **39.14 USD**
- Luka: ≈ **38.61 USD**

---

### Final Conclusion

After converting all wages to USD and calculating daily earnings, **Navin earns the most**.

$$
\boxed{Navin}
$$

[Assistant]
Explanation:
Okay, so Navin, Luka, and Ian are talking about their part-time jobs, and we need to figure out who earns the most after working an eight-hour day. The problem gives their hourly wages in their local currencies: Navin makes 160 rupee per hour, Luka makes 25 kuna per hour, and Ian makes 34 pula per hour. Then it gives the exchange rates for US dollars to each of their currencies. The question is, who earns the most after working 8 hours?

Alright, so first, I need to convert each of their hourly wages into US dollars, then multiply by 8 hours to get their daily earnings in USD. Then compare those amounts to see who has the highest. That seems like the plan. Let me write that down step by step.

First, let me note down the given information:

- Navin's wage: 160 rupee per hour
- Luka's wage: 25 kuna per hour
- Ian's wage: 34 pula per hour

Exchange rates:

- 1 USD = 32.35 Mauritius rupee
- 1 USD = 5.18 Croatian kuna
- 1 USD = 6.95 Botswana pula

So, to convert their wages to USD, I need to divide their hourly wage by the exchange rate. Because if 1 USD is equal to X local currency, then 1 local currency is equal to 1/X USD. For example, if 1 USD = 32.35 rupee, then 1 rupee is 1/32.35 USD. Therefore, Navin's hourly wage in USD would be 160 divided by 32.35. Similarly for the others.

Once I have their hourly wages in USD, multiply by 8 hours to get the total earnings for the day. Then compare the three totals.

Let me compute each one step by step.

Starting with Navin:

Navin's hourly wage in rupee: 160 rupee/hour

Exchange rate: 1 USD = 32.35 rupee

So, to convert to USD: 160 / 32.35

Let me calculate that. Let me get a calculator or do it manually. Let me approximate.

First, 32.35 times 5 is 161.75. Wait, 32.35 * 5 = 161.75. But Navin's wage is 160, which is slightly less than 161.75. So 160 / 32.35 is approximately 4.947... USD per hour? Wait, because 32.35 * 4.947 ≈ 160?

Wait, let me check:

Let me compute 160 divided by 32.35.

Let me write it as 160 / 32.35.

Let me divide numerator and denominator by 5 to simplify? Let me see:

32.35 divided by 5 is 6.47, and 160 divided by 5 is 32. So 32 / 6.47 ≈ ?

Well, 6.47 * 5 = 32.35, which is slightly more than 32. So 32 / 6.47 ≈ 4.947. So approximately 4.947 USD per hour. Let me check with actual division.

Alternatively, maybe I can use decimal division.

Alternatively, since 32.35 * 4.9 = ?

Let me compute 32.35 * 4.9:

First, 32.35 * 4 = 129.4

32.35 * 0.9 = 29.115

Adding together: 129.4 + 29.115 = 158.515

So 32.35 * 4.9 = 158.515

But Navin's wage is 160, which is 160 - 158.515 = 1.485 more.

So how much more is that? Since 32.35 * x = 1.485, so x ≈ 1.485 / 32.35 ≈ 0.0459

Therefore, total is approximately 4.9 + 0.0459 ≈ 4.9459 USD per hour. So approximately 4.946 USD per hour.

So Navin's hourly wage in USD is approximately 4.946 USD.

Therefore, for 8 hours, Navin earns 4.946 * 8.

Let me compute that. 4.946 * 8:

Well, 4 * 8 = 32, 0.946 * 8 = 7.568, so total is 32 + 7.568 = 39.568 USD. So approximately 39.57 USD per day.

Okay, that's Navin.

Now moving on to Luka.

Luka's wage: 25 kuna per hour.

Exchange rate: 1 USD = 5.18 kuna.

Therefore, to convert to USD: 25 / 5.18.

Let me compute that. Let me see.

First, 5.18 * 4 = 20.72

Subtract that from 25: 25 - 20.72 = 4.28

So that's 4 + (4.28 / 5.18). Let me compute 4.28 / 5.18.

Well, 5.18 * 0.8 = 4.144

So 4.28 - 4.144 = 0.136

So that's 0.8 + (0.136 / 5.18). 0.136 / 5.18 ≈ 0.0263

So total is approximately 4.8 + 0.0263 ≈ 4.8263 USD per hour.

Wait, let me check again:

Wait, original value is 25 / 5.18. Let me do this division properly.

Alternatively, approximate decimal.

Let me note that 5.18 * 4.8 = ?

Well, 5 * 4.8 = 24, 0.18 * 4.8 = 0.864, so total is 24 + 0.864 = 24.864

Which is less than 25. The difference is 25 - 24.864 = 0.136.

So, as before, so 4.8 + (0.136 / 5.18). Which is approximately 4.8 + 0.0263 ≈ 4.8263 USD per hour.

So approximately 4.826 USD per hour.

Therefore, for 8 hours, Luka earns 4.826 * 8.

Let me compute that. 4 * 8 = 32, 0.826 * 8 = 6.608, so total is 32 + 6.608 = 38.608 USD. Approximately 38.61 USD per day.

So Luka earns about 38.61 USD per day.

Now Ian.

Ian's wage: 34 pula per hour.

Exchange rate: 1 USD = 6.95 pula.

Therefore, converting to USD: 34 / 6.95.

Let me compute that. Let me see.

First, 6.95 * 4 = 27.8

Subtract that from 34: 34 - 27.8 = 6.2

So that's 4 + (6.2 / 6.95). Let me compute 6.2 / 6.95.

Well, 6.95 * 0.89 ≈ ?

Wait, 6.95 * 0.9 = 6.255. Which is slightly more than 6.2. So 0.9 - (6.255 - 6.2)/6.95.

Wait, 6.255 - 6.2 = 0.055. So 0.055 / 6.95 ≈ 0.0079. Therefore, approximately 0.9 - 0.0079 ≈ 0.8921.

Therefore, total is approximately 4 + 0.8921 ≈ 4.8921 USD per hour.

Alternatively, let me check with another method.

Let me compute 34 divided by 6.95.

Let me write this as 34 / 6.95. Multiply numerator and denominator by 100 to eliminate decimals: 3400 / 695.

Simplify this fraction. Let me divide numerator and denominator by 5: 3400 / 5 = 680, 695 / 5 = 139. So 680 / 139.

Now, let me divide 680 by 139.

139 * 4 = 556

Subtract from 680: 680 - 556 = 124

Bring down a zero (making it 1240). 139 * 8 = 1112

Subtract: 1240 - 1112 = 128

Bring down another zero: 1280. 139 * 9 = 1251

Subtract: 1280 - 1251 = 29

So so far, we have 4.89... So approximately 4.89 USD per hour.

So approximately 4.89 USD per hour.

Therefore, for 8 hours, Ian earns 4.89 * 8.

Let me compute that. 4 * 8 = 32, 0.89 * 8 = 7.12, so total is 32 + 7.12 = 39.12 USD per day.

So Ian earns approximately 39.12 USD per day.

So summarizing:

Navin: ~39.57 USD

Luka: ~38.61 USD

Ian: ~39.12 USD

Therefore, Navin earns the most, followed by Ian, then Luka.

Wait, but let me check my calculations again because approximations can sometimes lead to errors. Let me verify each calculation more precisely.

Starting with Navin:

160 rupee per hour. 1 USD = 32.35 rupee. So 160 / 32.35.

Let me compute this division more accurately.

Let me use decimal division.

Let me write 160 ÷ 32.35.

First, note that 32.35 × 4 = 129.4

Subtract that from 160: 160 - 129.4 = 30.6

Bring down a decimal: So we have 30.6 divided by 32.35. Let me add decimal places.

So 30.6 / 32.35 = 0.945... Wait, but actually, since we have 4.??? So total is 4 + (30.6 / 32.35). Let me compute 30.6 / 32.35.

Multiply numerator and denominator by 100: 3060 / 3235.

Divide numerator and denominator by 5: 612 / 647 ≈ ?

Well, 647 × 0.945 ≈ ?

Wait, maybe another approach. Let me approximate 30.6 / 32.35 ≈ 0.945. Wait, because 32.35 × 0.945 ≈ 32.35 × 0.9 + 32.35 × 0.045 = 29.115 + 1.45575 ≈ 30.57075. Which is very close to 30.6. Therefore, 30.6 / 32.35 ≈ 0.945. Therefore, total is 4.945 USD per hour. So 4.945 * 8 = ?

4.945 * 8:

Well, 4 * 8 = 32, 0.945 * 8 = 7.56, so total is 32 + 7.56 = 39.56 USD. So Navin's total is approximately 39.56 USD.

Earlier I had 39.57, which is consistent.

Now Luka: 25 kuna per hour. 1 USD = 5.18 kuna. So 25 / 5.18.

Let me compute this division more accurately.

Let me do 25 ÷ 5.18.

First, 5.18 × 4 = 20.72

Subtract from 25: 25 - 20.72 = 4.28

Bring decimal places. So now, 4.28 / 5.18.

Multiply numerator and denominator by 100: 428 / 518.

Divide numerator and denominator by 2: 214 / 259 ≈ ?

Let me compute 214 ÷ 259.

Well, 259 × 0.8 = 207.2

Subtract from 214: 214 - 207.2 = 6.8

So that's 0.8 + (6.8 / 259). 6.8 / 259 ≈ 0.02625.

Therefore, total is approximately 0.82625. Therefore, total value is 4 + 0.82625 ≈ 4.82625 USD per hour.

Therefore, 4.82625 * 8 = ?

4 * 8 = 32, 0.82625 * 8 = 6.61, so total is 32 + 6.61 = 38.61 USD. Which matches my previous calculation.

Now Ian: 34 pula per hour. 1 USD = 6.95 pula. So 34 / 6.95.

Let me compute this division more accurately.

34 ÷ 6.95.

First, 6.95 × 4 = 27.8

Subtract from 34: 34 - 27.8 = 6.2

Bring decimal places. So 6.2 / 6.95.

Multiply numerator and denominator by 100: 620 / 695.

Divide numerator and denominator by 5: 124 / 139 ≈ ?

Let me compute 124 ÷ 139.

Well, 139 × 0.89 = ?

139 × 0.8 = 111.2

139 × 0.09 = 12.51

Total: 111.2 + 12.51 = 123.71

Which is very close to 124. The difference is 124 - 123.71 = 0.29.

So 0.89 + (0.29 / 139) ≈ 0.89 + 0.00208 ≈ 0.89208.

Therefore, total value is 4 + 0.89208 ≈ 4.89208 USD per hour.

Therefore, 4.89208 * 8 = ?

4 * 8 = 32, 0.89208 * 8 = ?

Well, 0.8 * 8 = 6.4, 0.09208 * 8 ≈ 0.73664. So total is 6.4 + 0.73664 ≈ 7.13664. Therefore, total earnings: 32 + 7.13664 ≈ 39.13664 USD. Approximately 39.14 USD.

So Ian earns approximately 39.14 USD per day.

So comparing:

Navin: ~39.56 USD

Ian: ~39.14 USD

Luka: ~38.61 USD

Therefore, Navin earns the most.

But let me check once again if there's any miscalculations. Let me verify with exact decimal computations or perhaps use fractions.

Alternatively, maybe I can use calculator-like computations.

Alternatively, let me check Navin's rate again.

Navin: 160 / 32.35.

Let me compute 160 / 32.35.

Let me note that 32.35 × 4.94 = ?

Well, 32.35 × 4 = 129.4

32.35 × 0.9 = 29.115

32.35 × 0.04 = 1.294

So total for 4.94 is 129.4 + 29.115 + 1.294 = 129.4 + 30.409 = 159.809

Which is very close to 160. The difference is 160 - 159.809 = 0.191.

So, 0.191 / 32.35 ≈ 0.0059. Therefore, total is 4.94 + 0.0059 ≈ 4.9459 USD per hour. So 4.9459 * 8 = 39.5672 USD. So approximately 39.57 USD. Correct.

For Ian: 34 / 6.95.

Let me compute 34 / 6.95.

Let me note that 6.95 × 4.89 = ?

Well, 6.95 × 4 = 27.8

6.95 × 0.8 = 5.56

6.95 × 0.09 = 0.6255

Adding those: 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is very close to 34. The difference is 34 - 33.9855 = 0.0145.

So, 0.0145 / 6.95 ≈ 0.002086. Therefore, total is 4.89 + 0.002086 ≈ 4.892086 USD per hour. Multiply by 8: 4.892086 * 8 ≈ 39.1367 USD. So approximately 39.14 USD. Correct.

So Navin is at ~39.57, Ian ~39.14, Luka ~38.61. Therefore, Navin earns the most.

But let me check if there's any chance that my approximated decimal calculations might have errors. For example, maybe I should carry out more precise calculations or use fractions?

Alternatively, perhaps I can use exact fractions with the given exchange rates.

But given that the exchange rates are given to two decimal places, maybe the question expects us to use those exact rates and compute accordingly. Let me check if there's any possibility that rounding during the calculation might affect the result.

Alternatively, maybe I can compute each value with more precision.

Let me try Navin again:

Navin's hourly wage in USD: 160 / 32.35

Let me compute this division with more precision.

Let me write 160 ÷ 32.35.

Let me convert this to 16000 ÷ 3235 (multiplying numerator and denominator by 100 to eliminate decimals).

Now, let me perform the division 16000 ÷ 3235.

First, 3235 × 4 = 12940

Subtract from 16000: 16000 - 12940 = 3060

Bring down a zero (making it 30600). Now divide 30600 by 3235.

3235 × 9 = 29115

Subtract from 30600: 30600 - 29115 = 1485

Bring down a zero (making it 14850). Divide by 3235.

3235 × 4 = 12940

Subtract: 14850 - 12940 = 1910

Bring down a zero (making it 19100). Divide by 3235.

3235 × 5 = 16175

Subtract: 19100 - 16175 = 2925

Bring down a zero (making it 29250). Divide by 3235.

3235 × 9 = 29115

Subtract: 29250 - 29115 = 135

So so far, we have 4.9459... So the value is approximately 4.9459 USD per hour. Therefore, 4.9459 * 8 = ?

Let me compute 4.9459 * 8:

First, 4 * 8 = 32

0.9459 * 8:

0.9 * 8 = 7.2

0.0459 * 8 = 0.3672

Total: 7.2 + 0.3672 = 7.5672

Therefore, total earnings: 32 + 7.5672 = 39.5672 USD. So approximately 39.57 USD. Correct.

For Ian: 34 / 6.95.

Let me compute 34 ÷ 6.95.

Convert to 3400 ÷ 695.

Divide numerator and denominator by 5: 680 ÷ 139.

Let me perform this division.

139 × 4 = 556

Subtract from 680: 680 - 556 = 124

Bring down a zero: 1240

139 × 8 = 1112

Subtract: 1240 - 1112 = 128

Bring down a zero: 1280

139 × 9 = 1251

Subtract: 1280 - 1251 = 29

Bring down a zero: 290

139 × 2 = 278

Subtract: 290 - 278 = 12

Bring down a zero: 120

139 × 0 = 0

So so far, we have 4.892... So the value is approximately 4.892 USD per hour. Multiply by 8:

4.892 * 8 = ?

Well, 4 * 8 = 32, 0.892 * 8 = 7.136, so total is 32 + 7.136 = 39.136 USD. So approximately 39.14 USD.

Therefore, Navin is at ~39.57, Ian ~39.14, Luka ~38.61. So Navin is the highest.

But let me check once again for Luka, just to be thorough.

Luka: 25 / 5.18.

Convert to 2500 / 518.

Divide numerator and denominator by 2: 1250 / 259.

Let me perform this division.

259 × 4 = 1036

Subtract from 1250: 1250 - 1036 = 214

Bring down a zero: 2140

259 × 8 = 2072

Subtract: 2140 - 2072 = 68

Bring down a zero: 680

259 × 2 = 518

Subtract: 680 - 518 = 162

Bring down a zero: 1620

259 × 6 = 1554

Subtract: 1620 - 1554 = 66

Bring down a zero: 660

259 × 2 = 518

Subtract: 660 - 518 = 142

So so far, we have 4.826... So the value is approximately 4.826 USD per hour. Multiply by 8:

4.826 * 8 = ?

4 * 8 = 32, 0.826 * 8 = 6.608, total 38.608 USD. So ~38.61 USD. Correct.

Therefore, the calculations are consistent.

Therefore, Navin earns the most, followed by Ian, then Luka.

But let me check if there's any chance that the question expects us to consider something else, like maybe the exchange rates are given as USD to local currency, so maybe we need to check if the question is about converting their earnings to USD or comparing in their own currencies? Wait, no, the question says "who would earn the most after working an eight-hour day?" and since they are discussing their part-time jobs, but the question is likely asking in USD terms because the exchange rates are given. Otherwise, if we compare in their own currencies, we can't compare because they are different currencies. So the question must be asking for the earnings converted to USD.

Alternatively, maybe the question is asking in their own currencies? But that wouldn't make sense because the currencies are different. So the answer must be in USD.

Alternatively, maybe the question is asking who earns the most in their own currency? But that would be impossible to compare unless we have exchange rates. But since exchange rates are given, it's more likely that we need to convert to USD.

Alternatively, maybe the question is asking for the total earnings in their own currency? But then, since they are different currencies, we can't compare. Therefore, the only logical way is to convert to USD.

Therefore, the answer is Navin.

But let me check once again if I did the calculations correctly. Maybe I made a mistake in the exchange rates?

Wait, the exchange rates are:

1 USD = 32.35 Mauritius rupee

So to get rupee to USD, divide by 32.35. Correct.

Similarly for others. Correct.

So, yes, the calculations are correct.

Therefore, Navin earns the most.

But let me check once again with exact decimal values.

Alternatively, maybe I can use calculator-like computations for each.

For Navin:

160 / 32.35 = ?

Let me approximate 32.35 × 4.94 = ?

As before, 32.35 × 4.94 ≈ 159.8, which is very close to 160. So 4.94 USD per hour. So 4.94 × 8 = 39.52 USD. But earlier precise calculation gave 39.5672. So depending on rounding, maybe 39.57.

For Ian:

34 / 6.95 ≈ 4.892 USD per hour. 4.892 × 8 = 39.136 USD.

So Navin is higher.

Therefore, the conclusion is Navin earns the most.

But let me check if there's any chance that the question is in another way. For example, maybe the exchange rates are given as USD to local currency, so to find the earnings in USD, we need to multiply their local currency earnings by the exchange rate? Wait, no. Wait, if 1 USD = X local currency, then 1 local currency = 1/X USD. Therefore, to convert local currency to USD, you divide by X. So for example, if you have Y local currency, then in USD it's Y / X.

Alternatively, if the exchange rate was given as local currency per USD, then you would multiply. But the question says "one US dollar is equivalent to 32.35 Mauritius rupee", meaning 1 USD = 32.35 rupee. Therefore, to get USD from rupee, divide by 32.35. Correct.

Therefore, calculations are correct.

Therefore, the answer is Navin.

But let me check once again with exact numbers.

Navin: 160 / 32.35 = ?

Let me compute 160 / 32.35:

Let me use cross multiplication.

Let me consider that 32.35 × 4.94 = ?

As above, 32.35 × 4.94 = 32.35 × (4 + 0.9 + 0.04) = 32.35×4 + 32.35×0.9 + 32.35×0.04

= 129.4 + 29.115 + 1.294 = 129.4 + 29.115 = 158.515 + 1.294 = 159.809

Which is 159.809, which is 0.191 less than 160. So, to get the exact value, we can write:

Let me denote x = 160 / 32.35

We have 32.35 × x = 160

We know that 32.35 × 4.94 = 159.809

So, 32.35 × (4.94 + Δ) = 160

So, 32.35 × Δ = 160 - 159.809 = 0.191

Therefore, Δ = 0.191 / 32.35 ≈ 0.0059

Therefore, x ≈ 4.94 + 0.0059 ≈ 4.9459 USD per hour.

Therefore, 4.9459 × 8 = 39.5672 USD.

Similarly, for Ian:

34 / 6.95 = ?

Let me compute 6.95 × 4.89 = ?

As above, 6.95 × 4.89 = 6.95 × (4 + 0.8 + 0.09) = 6.95×4 + 6.95×0.8 + 6.95×0.09

= 27.8 + 5.56 + 0.6255 = 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is 33.9855, which is 0.0145 less than 34. Therefore, Δ = 0.0145 / 6.95 ≈ 0.002086

Therefore, x ≈ 4.89 + 0.002086 ≈ 4.892086 USD per hour.

Multiply by 8: 4.892086 × 8 = 39.1367 USD.

So, yes, Navin is higher.

Therefore, the answer is Navin.

But let me check once again if there's any possible mistake in the exchange rates. For example, if I confused the exchange rates?

Wait, the problem says:

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

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

Similarly, 1 USD = 5.18 kuna, so kuna to USD is divide by 5.18. Correct.

Same for pula. Correct.

Therefore, calculations are correct.

Therefore, Navin earns the most.

But just to make sure, let me check with another approach. Let me compute all three in USD:

Navin: 160 / 32.35 ≈ 4.9459 USD/hour

Luka: 25 / 5.18 ≈ 4.8263 USD/hour

Ian: 34 / 6.95 ≈ 4.8921 USD/hour

Then multiply by 8:

Navin: ~39.57 USD

Luka: ~38.61 USD

Ian: ~39.14 USD

Yes, so Navin is the highest.

Therefore, the answer is Navin.

**Final Answer**
\boxed{Navin}

To determine who earns the most after working an eight-hour day, we need to convert each person's hourly wage from their local currency to US dollars (USD) using the given exchange rates. Then, we multiply each hourly wage in USD by 8 to find the total daily earnings in USD. Finally, we compare the results.

---

### Step 1: Convert Hourly Wages to USD

#### Navin (Mauritius Rupee)
- Hourly wage: 160 rupee
- Exchange rate: 1 USD = 32.35 rupee
- USD per hour:
  $$
  \frac{160}{32.35} \approx 4.9459 \text{ USD}
  $$

#### Luka (Croatian Kuna)
- Hourly wage: 25 kuna
- Exchange rate: 1 USD = 5.18 kuna
- USD per hour:
  $$
  \frac{25}{5.18} \approx 4.8263 \text{ USD}
  $$

#### Ian (Botswana Pula)
- Hourly wage: 34 pula
- Exchange rate: 1 USD = 6.95 pula
- USD per hour:
  $$
  \frac{34}{6.95} \approx 4.8921 \text{ USD}
  $$

---

### Step 2: Calculate Daily Earnings (8 hours)

#### Navin
$$
4.9459 \times 8 \approx 39.57 \text{ USD}
$$

#### Luka
$$
4.8263 \times 8 \approx 38.61 \text{ USD}
$$

#### Ian
$$
4.8921 \times 8 \approx 39.14 \text{ USD}
$$

---

### Step 3: Compare Earnings

- Navin: ≈ **39.57 USD**
- Ian: ≈ **39.14 USD**
- Luka: ≈ **38.61 USD**

---

### Final Conclusion

After converting all wages to USD and calculating daily earnings, **Navin earns the most**.

$$
\boxed{Navin}
$$

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.