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, so the numerator and denominator must have opposite signs. Let me write that down.

The inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. Since the absolute value in the numerator is always greater than or equal to zero, the sign of the entire expression depends on the denominator. However, since the absolute value is non-negative, the fraction will be negative only if the denominator is negative. Wait, but if the numerator is zero, then the whole expression is zero, which isn't less than zero. So, the numerator can't be zero. Therefore, the inequality is equivalent to the denominator being negative, and the numerator not being zero. But let me check that again.

Wait, actually, the absolute value is always non-negative, so the numerator is non-negative. Therefore, the fraction is negative only when the denominator is negative. Because if the numerator is positive and the denominator is negative, the whole fraction is negative. If the numerator is zero, the fraction is zero, which doesn't satisfy the inequality. So, the inequality reduces to two conditions:

1. $x^2 - 36x < 0$
2. $|x^2 - 81| \neq 0$

But let me confirm. If the numerator is zero, then the fraction is zero, which is not less than zero. Therefore, we need to exclude the points where the numerator is zero. So, first, let me find where the numerator is zero. The numerator is $|x^2 - 81|$, which is zero when $x^2 - 81 = 0$, so $x^2 = 81$, so $x = 9$ or $x = -9$. Therefore, we need to exclude $x = 9$ and $x = -9$ from the solution set.

Now, moving on to the denominator: $x^2 - 36x < 0$. Let me factor that. $x^2 - 36x = x(x - 36)$. So, the denominator is negative when $x(x - 36) < 0$. To solve this inequality, I can find the critical points where the expression equals zero, which are $x = 0$ and $x = 36$. Then, I can test intervals between these critical points.

So, the critical points divide the real number line into three intervals:

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

Let me test each interval.

For $x < 0$: Let's pick $x = -1$. Then, $x = -1$ is negative, $x - 36 = -37$ is also negative. Multiplying two negatives gives a positive. So, $x(x - 36) > 0$ here. Therefore, this interval does not satisfy the inequality.

For $0 < x < 36$: Let's pick $x = 18$. Then, $x = 18$ is positive, $x - 36 = -18$ is negative. Multiplying positive and negative gives negative. Therefore, $x(x - 36) < 0$ here. So, this interval satisfies the inequality.

For $x > 36$: Let's pick $x = 40$. Then, $x = 40$ is positive, $x - 36 = 4$ is positive. Multiplying two positives gives positive. Therefore, $x(x - 36) > 0$ here. So, this interval does not satisfy the inequality.

Therefore, the denominator is negative when $0 < x < 36$. However, we also need to exclude the points where the numerator is zero, which are $x = 9$ and $x = -9$. Wait, but $x = -9$ is not in the interval $0 < x < 36$, so we don't need to exclude it from this interval. However, $x = 9$ is in the interval $0 < x < 36$, so we need to exclude $x = 9$ from the solution set.

Therefore, the solution set is $0 < x < 36$ excluding $x = 9$. But let me check if there are any other restrictions. Also, we need to make sure that the denominator isn't zero. The denominator is zero when $x = 0$ or $x = 36$, so those points are excluded as well. Therefore, the original expression is undefined at $x = 0$ and $x = 36$, so those points are excluded from the domain.

Therefore, combining all that, the solution is $0 < x < 36$ except $x = 9$. But let me check if there are any other points where the numerator is zero. The numerator is zero at $x = 9$ and $x = -9$, but $x = -9$ is not in the interval $0 < x < 36$, so we only need to exclude $x = 9$.

Wait, but hold on. Let me think again. The original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. So, even though the numerator is non-negative, the denominator must be negative. However, is there a case where the numerator is zero? If the numerator is zero, then the entire expression is zero, which is not less than zero. Therefore, we need to exclude those points where the numerator is zero. So, as we said, $x = 9$ and $x = -9$. However, $x = -9$ is not in the interval where the denominator is negative (which is $0 < x < 36$), so we don't have to worry about excluding it from that interval. Therefore, the solution set is $0 < x < 36$ excluding $x = 9$.

But let me check if there are any other points where the denominator is negative. Wait, the denominator is negative only in $0 < x < 36$, so that's the only interval. Therefore, the solution is $0 < x < 36$ except $x = 9$. However, let me check if there are any other restrictions. For example, maybe the absolute value can affect the sign? Wait, no. The absolute value is always non-negative, so the sign of the entire expression is determined by the denominator. Therefore, if the denominator is negative, the entire expression is negative (since numerator is non-negative, but not zero). Therefore, the solution is indeed $0 < x < 36$ excluding $x = 9$.

But let me test some values to make sure. Let's take $x = 1$, which is in the interval $0 < x < 36$ and not equal to 9. Then, compute the expression:

Numerator: $|1^2 - 81| = |1 - 81| = | -80 | = 80$.

Denominator: $1^2 - 36*1 = 1 - 36 = -35$.

So, the fraction is 80 / (-35) = -80/35 < 0. So, that works.

Now, take $x = 9$. The numerator is |81 - 81| = 0, so the expression is 0, which is not less than 0. So, excluded.

Take $x = 36$. The denominator is 36^2 - 36*36 = 0, so undefined. Excluded.

Take $x = 0$. Denominator is 0, undefined. Excluded.

Take $x = -1$. The denominator is (-1)^2 - 36*(-1) = 1 + 36 = 37 > 0. So, the fraction is |...| / positive, which is non-negative. So, not less than 0. So, not in the solution set. Correct.

Take $x = 10$. Let's check. Numerator: |100 - 81| = 19. Denominator: 100 - 360 = -260. So, 19 / (-260) < 0. Correct.

Take $x = 35$. Numerator: |35^2 - 81| = |1225 - 81| = 1144. Denominator: 35^2 - 36*35 = 1225 - 1260 = -35. So, 1144 / (-35) < 0. Correct.

Take $x = 9$. As before, 0. Not included.

Take $x = 5$. Numerator: |25 - 81| = 56. Denominator: 25 - 180 = -155. So, negative. Correct.

So, seems like the solution is $0 < x < 36$ except $x = 9$. Therefore, in interval notation, that would be $(0, 9) \cup (9, 36)$.

Wait, but hold on. Let me check if there are any other points where the numerator is zero. For example, 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 / positive = 0, which is not less than zero. Therefore, even though the numerator is zero, the expression is zero, so it's excluded. However, since $x = -9$ is not in the interval where the denominator is negative, we don't have to worry about it here. Therefore, the solution set is indeed $(0, 9) \cup (9, 36)$.

But let me think again. Is there a case where the absolute value could make the numerator positive or negative? Wait, no. The absolute value is always non-negative. So, the numerator is always non-negative, and the denominator is negative in the interval $0 < x < 36$. Therefore, the entire expression is negative except when the numerator is zero. Therefore, the solution is $0 < x < 36$ excluding $x = 9$.

But let me check if there's a possibility that the absolute value could be zero in the interval $0 < x < 36$. For example, $x = 9$ is in that interval, and that's where the numerator is zero. So, yes, only at $x = 9$. Therefore, the solution is $0 < x < 36$ except $x = 9$.

But let me also 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. Therefore, the final answer is $0 < x < 36$ with $x \neq 9$.

But let me write that in interval notation. So, from 0 to 9, not including 0 and 9, and from 9 to 36, not including 9 and 36. Therefore, $(0, 9) \cup (9, 36)$.

But let me check if there's a mistake here. Suppose someone might think that the absolute value could affect the sign. For example, if the expression inside the absolute value is negative, then the absolute value would make it positive. But since the absolute value is always non-negative, regardless of the inside, the numerator is non-negative. Therefore, the sign of the entire expression is determined by the denominator. Therefore, the only time the expression is negative is when the denominator is negative and the numerator is not zero. Therefore, the solution is as above.

Alternatively, maybe I can approach this problem by considering cases based on the absolute value. Let me try that approach to verify.

The absolute value $|x^2 - 81|$ can be split into two cases:

Case 1: $x^2 - 81 \geq 0$, which implies $x^2 \geq 81$, so $x \geq 9$ or $x \leq -9$.

Case 2: $x^2 - 81 < 0$, which implies $x^2 < 81$, so $-9 < x < 9$.

So, in Case 1: $x^2 - 81 \geq 0$, so $|x^2 - 81| = x^2 - 81$. Therefore, the inequality becomes $\frac{x^2 - 81}{x^2 - 36x} < 0$.

In Case 2: $x^2 - 81 < 0$, so $|x^2 - 81| = -(x^2 - 81) = -x^2 + 81$. Therefore, the inequality becomes $\frac{-x^2 + 81}{x^2 - 36x} < 0$.

So, let me analyze both cases separately.

Starting with Case 1: $x \geq 9$ or $x \leq -9$, and $\frac{x^2 - 81}{x^2 - 36x} < 0$.

First, note that in this case, the numerator is $x^2 - 81$, which is non-negative, and the denominator is $x^2 - 36x$. So, the fraction is negative when the denominator is negative (since numerator is non-negative). Therefore, similar to before, we need $x^2 - 36x < 0$, which is $0 < x < 36$. However, in Case 1, we have $x \geq 9$ or $x \leq -9$. Therefore, the overlap between $x \geq 9$ or $x \leq -9$ and $0 < x < 36$ is $9 \leq x < 36$. However, we also need to remember that in Case 1, the numerator is non-negative, but we need the fraction to be negative, so denominator must be negative. Therefore, the solution in Case 1 is $9 \leq x < 36$. However, we also need to check if the numerator is zero here. The numerator is zero when $x = 9$ or $x = -9$. But in this case, $x \geq 9$ or $x \leq -9$, so $x = 9$ is included here. However, at $x = 9$, the numerator is zero, so the fraction is zero, which is not less than zero. Therefore, we need to exclude $x = 9$. Therefore, in Case 1, the solution is $9 < x < 36$.

Now, moving to Case 2: $-9 < x < 9$, and the inequality is $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me simplify this. The numerator is $-x^2 + 81 = -(x^2 - 81)$. The denominator is $x^2 - 36x$. So, the inequality becomes $\frac{-(x^2 - 81)}{x^2 - 36x} < 0$, which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$ (multiplying numerator and denominator by -1 reverses the inequality sign). Wait, let me check that again.

Original inequality in Case 2: $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me factor numerator and denominator:

Numerator: $-x^2 + 81 = -(x^2 - 81) = -(x - 9)(x + 9)$.

Denominator: $x^2 - 36x = x(x - 36)$.

So, the inequality becomes $\frac{ - (x - 9)(x + 9) }{x(x - 36)} < 0$. Which is equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ (multiplying both sides by -1 reverses the inequality).

So, now we have $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. Let me analyze this inequality. The critical points are where the numerator or denominator is zero: $x = -9$, $x = 0$, $x = 9$, $x = 36$. These divide the real line into intervals:

1. $x < -9$
2. $-9 < x < 0$
3. $0 < x < 9$
4. $9 < x < 36$
5. $x > 36$

But remember, in Case 2, we have $-9 < x < 9$. Therefore, we only need to consider intervals 2, 3, and 4 (but 4 is part of Case 1? Wait, no. Wait, Case 2 is $-9 < x < 9$, so in this case, the intervals to consider are:

- $-9 < x < 0$
- $0 < x < 9$

Because $x = -9$ and $x = 9$ are excluded from Case 2 (since Case 2 is $-9 < x < 9$). Also, $x = 0$ is excluded because denominator is zero there.

So, let me check the sign of $\frac{(x - 9)(x + 9)}{x(x - 36)}$ in these intervals.

First interval: $-9 < x < 0$.

Let me pick $x = -1$:

Numerator: $(-1 - 9)(-1 + 9) = (-10)(8) = -80$

Denominator: $(-1)(-1 - 36) = (-1)(-37) = 37$

So, the fraction is $-80 / 37 < 0$. But we have the inequality $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. Therefore, this interval does not satisfy the inequality.

Second interval: $0 < x < 9$. Let me pick $x = 1$:

Numerator: $(1 - 9)(1 + 9) = (-8)(10) = -80$

Denominator: $1(1 - 36) = 1(-35) = -35$

So, the fraction is $-80 / -35 = 80/35 > 0$. Therefore, this interval satisfies the inequality.

Third interval: $x = 9$ is excluded. So, in Case 2, we only have $-9 < x < 9$, so the only interval that satisfies the inequality in Case 2 is $0 < x < 9$.

Therefore, combining Case 1 and Case 2:

Case 1 gives $9 < x < 36$.

Case 2 gives $0 < x < 9$.

Therefore, the total solution is $0 < x < 9$ or $9 < x < 36$, which is the same as $(0, 9) \cup (9, 36)$. Therefore, confirming the previous result.

But let me check if there are any other points. For example, in Case 2, when we had $-9 < x < 0$, the inequality was not satisfied. So, nothing there. Also, in Case 1, we had $9 < x < 36$. So, combining both cases, the solution is indeed $(0, 9) \cup (9, 36)$.

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

But let me check if there are any other restrictions. For example, in Case 2, when we had $-9 < x < 9$, but the original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. However, in Case 2, we transformed the inequality into $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$, which led us to $0 < x < 9$. But in this interval, the denominator $x^2 - 36x$ is negative (as we saw earlier), and the numerator $|x^2 - 81|$ is positive (since $x^2 - 81 < 0$ in this interval, so absolute value makes it positive). Therefore, the fraction is positive divided by negative, which is negative. Wait, but in Case 2, we had transformed the inequality into $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$, which is equivalent to the original inequality. Wait, but in Case 2, the original inequality is $\frac{-x^2 + 81}{x^2 - 36x} < 0$, which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$. But in the interval $0 < x < 9$, the numerator $x^2 - 81$ is negative, and the denominator $x^2 - 36x$ is negative. Therefore, negative divided by negative is positive, which satisfies the inequality $\frac{x^2 - 81}{x^2 - 36x} > 0$. Therefore, the original inequality $\frac{-x^2 + 81}{x^2 - 36x} < 0$ is equivalent to this. Therefore, in this interval, the original inequality is satisfied. Therefore, the solution in Case 2 is $0 < x < 9$, and in Case 1, $9 < x < 36$. Therefore, combining both, the solution is $0 < x < 36$ excluding $x = 9$.

Therefore, the answer is $(0, 9) \cup (9, 36)$.

But let me check if there's a mistake in the sign when I transformed the inequality. Let me re-express:

Original inequality in Case 2: $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me write this as $\frac{-(x^2 - 81)}{x^2 - 36x} < 0$. Which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$ because multiplying numerator and denominator by -1 reverses the inequality. Therefore, yes, that's correct. Therefore, solving $\frac{x^2 - 81}{x^2 - 36x} > 0$ in the interval $-9 < x < 9$.

But in this interval, $x^2 - 81 < 0$ and $x^2 - 36x < 0$ (since $0 < x < 36$), so the fraction is negative divided by negative, which is positive. Therefore, the inequality is satisfied. Therefore, the solution in Case 2 is $0 < x < 9$ (since $-9 < x < 0$ gives negative over positive, which is negative, not satisfying the inequality). Therefore, correct.

Therefore, combining both cases, the solution is indeed $0 < x < 9$ and $9 < x < 36$.

Therefore, the final answer is $(0, 9) \cup (9, 36)$.

But let me check if there are any other points where the original expression is defined. For example, at $x = 0$ and $x = 36$, the denominator is zero, so they are excluded. At $x = 9$, the numerator is zero, so the expression is zero, which is not less than zero, so excluded. At $x = -9$, the numerator is zero, but the denominator is positive, so the expression is zero, which is not less than zero, so excluded. Therefore, all other points in the intervals are okay.

Therefore, I think this is correct. Therefore, the solution is $0 < x < 36$ excluding $x = 9$, which is $(0, 9) \cup (9, 36)$.

**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 analyze the behavior of the expression by considering the sign of the numerator and denominator.

---

### Step 1: Understand the Structure

The expression is a rational function with an **absolute value** in the numerator and a **quadratic expression** in the denominator. The absolute value ensures the numerator is **non-negative** (i.e., $ |x^2 - 81| \geq 0 $), and the denominator is a quadratic expression that can be factored as:

$$
x^2 - 36x = x(x - 36).
$$

So, the sign of the entire expression depends on the **sign of the denominator**, since the numerator is always non-negative. The inequality is strict: the expression must be **less than zero**, which only occurs when the **numerator is positive** and the **denominator is negative**.

---

### Step 2: Determine When the Numerator is Zero

The numerator is $ |x^2 - 81| $, which is zero when:

$$
x^2 - 81 = 0 \Rightarrow x = \pm 9.
$$

At these points, the entire expression becomes zero, which **does not satisfy** the inequality $ < 0 $. So, we **exclude** $ x = 9 $ and $ x = -9 $ from the solution set.

---

### Step 3: Determine When the Denominator is Negative

We solve the inequality:

$$
x(x - 36) < 0.
$$

This is a quadratic inequality. The critical points are $ x = 0 $ and $ x = 36 $. Testing intervals:

- $ x < 0 $: $ x $ and $ x - 36 $ are both negative → product is **positive**
- $ 0 < x < 36 $: $ x $ is positive, $ x - 36 $ is negative → product is **negative**
- $ x > 36 $: both are positive → product is **positive**

So, the denominator is **negative** when $ 0 < x < 36 $.

---

### Step 4: Combine Conditions

We now combine the two conditions:

- The **denominator is negative** when $ 0 < x < 36 $
- The **numerator is not zero** when $ x \neq \pm 9 $

So, the expression is **negative** (i.e., satisfies the inequality) when:

$$
0 < x < 36 \quad \text{and} \quad x \neq 9.
$$

This gives the solution set:

$$
(0, 9) \cup (9, 36).
$$

---

### 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, so the numerator and denominator must have opposite signs. Let me write that down.

The inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. Since the absolute value in the numerator is always greater than or equal to zero, the sign of the entire expression depends on the denominator. However, since the absolute value is non-negative, the fraction will be negative only if the denominator is negative. Wait, but if the numerator is zero, then the whole expression is zero, which isn't less than zero. So, the numerator can't be zero. Therefore, the inequality is equivalent to the denominator being negative, and the numerator not being zero. But let me check that again.

Wait, actually, the absolute value is always non-negative, so the numerator is non-negative. Therefore, the fraction is negative only when the denominator is negative. Because if the numerator is positive and the denominator is negative, the whole fraction is negative. If the numerator is zero, the fraction is zero, which doesn't satisfy the inequality. So, the inequality reduces to two conditions:

1. $x^2 - 36x < 0$
2. $|x^2 - 81| \neq 0$

But let me confirm. If the numerator is zero, then the fraction is zero, which is not less than zero. Therefore, we need to exclude the points where the numerator is zero. So, first, let me find where the numerator is zero. The numerator is $|x^2 - 81|$, which is zero when $x^2 - 81 = 0$, so $x^2 = 81$, so $x = 9$ or $x = -9$. Therefore, we need to exclude $x = 9$ and $x = -9$ from the solution set.

Now, moving on to the denominator: $x^2 - 36x < 0$. Let me factor that. $x^2 - 36x = x(x - 36)$. So, the denominator is negative when $x(x - 36) < 0$. To solve this inequality, I can find the critical points where the expression equals zero, which are $x = 0$ and $x = 36$. Then, I can test intervals between these critical points.

So, the critical points divide the real number line into three intervals:

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

Let me test each interval.

For $x < 0$: Let's pick $x = -1$. Then, $x = -1$ is negative, $x - 36 = -37$ is also negative. Multiplying two negatives gives a positive. So, $x(x - 36) > 0$ here. Therefore, this interval does not satisfy the inequality.

For $0 < x < 36$: Let's pick $x = 18$. Then, $x = 18$ is positive, $x - 36 = -18$ is negative. Multiplying positive and negative gives negative. Therefore, $x(x - 36) < 0$ here. So, this interval satisfies the inequality.

For $x > 36$: Let's pick $x = 40$. Then, $x = 40$ is positive, $x - 36 = 4$ is positive. Multiplying two positives gives positive. Therefore, $x(x - 36) > 0$ here. So, this interval does not satisfy the inequality.

Therefore, the denominator is negative when $0 < x < 36$. However, we also need to exclude the points where the numerator is zero, which are $x = 9$ and $x = -9$. Wait, but $x = -9$ is not in the interval $0 < x < 36$, so we don't need to exclude it from this interval. However, $x = 9$ is in the interval $0 < x < 36$, so we need to exclude $x = 9$ from the solution set.

Therefore, the solution set is $0 < x < 36$ excluding $x = 9$. But let me check if there are any other restrictions. Also, we need to make sure that the denominator isn't zero. The denominator is zero when $x = 0$ or $x = 36$, so those points are excluded as well. Therefore, the original expression is undefined at $x = 0$ and $x = 36$, so those points are excluded from the domain.

Therefore, combining all that, the solution is $0 < x < 36$ except $x = 9$. But let me check if there are any other points where the numerator is zero. The numerator is zero at $x = 9$ and $x = -9$, but $x = -9$ is not in the interval $0 < x < 36$, so we only need to exclude $x = 9$.

Wait, but hold on. Let me think again. The original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. So, even though the numerator is non-negative, the denominator must be negative. However, is there a case where the numerator is zero? If the numerator is zero, then the entire expression is zero, which is not less than zero. Therefore, we need to exclude those points where the numerator is zero. So, as we said, $x = 9$ and $x = -9$. However, $x = -9$ is not in the interval where the denominator is negative (which is $0 < x < 36$), so we don't have to worry about excluding it from that interval. Therefore, the solution set is $0 < x < 36$ excluding $x = 9$.

But let me check if there are any other points where the denominator is negative. Wait, the denominator is negative only in $0 < x < 36$, so that's the only interval. Therefore, the solution is $0 < x < 36$ except $x = 9$. However, let me check if there are any other restrictions. For example, maybe the absolute value can affect the sign? Wait, no. The absolute value is always non-negative, so the sign of the entire expression is determined by the denominator. Therefore, if the denominator is negative, the entire expression is negative (since numerator is non-negative, but not zero). Therefore, the solution is indeed $0 < x < 36$ excluding $x = 9$.

But let me test some values to make sure. Let's take $x = 1$, which is in the interval $0 < x < 36$ and not equal to 9. Then, compute the expression:

Numerator: $|1^2 - 81| = |1 - 81| = | -80 | = 80$.

Denominator: $1^2 - 36*1 = 1 - 36 = -35$.

So, the fraction is 80 / (-35) = -80/35 < 0. So, that works.

Now, take $x = 9$. The numerator is |81 - 81| = 0, so the expression is 0, which is not less than 0. So, excluded.

Take $x = 36$. The denominator is 36^2 - 36*36 = 0, so undefined. Excluded.

Take $x = 0$. Denominator is 0, undefined. Excluded.

Take $x = -1$. The denominator is (-1)^2 - 36*(-1) = 1 + 36 = 37 > 0. So, the fraction is |...| / positive, which is non-negative. So, not less than 0. So, not in the solution set. Correct.

Take $x = 10$. Let's check. Numerator: |100 - 81| = 19. Denominator: 100 - 360 = -260. So, 19 / (-260) < 0. Correct.

Take $x = 35$. Numerator: |35^2 - 81| = |1225 - 81| = 1144. Denominator: 35^2 - 36*35 = 1225 - 1260 = -35. So, 1144 / (-35) < 0. Correct.

Take $x = 9$. As before, 0. Not included.

Take $x = 5$. Numerator: |25 - 81| = 56. Denominator: 25 - 180 = -155. So, negative. Correct.

So, seems like the solution is $0 < x < 36$ except $x = 9$. Therefore, in interval notation, that would be $(0, 9) \cup (9, 36)$.

Wait, but hold on. Let me check if there are any other points where the numerator is zero. For example, 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 / positive = 0, which is not less than zero. Therefore, even though the numerator is zero, the expression is zero, so it's excluded. However, since $x = -9$ is not in the interval where the denominator is negative, we don't have to worry about it here. Therefore, the solution set is indeed $(0, 9) \cup (9, 36)$.

But let me think again. Is there a case where the absolute value could make the numerator positive or negative? Wait, no. The absolute value is always non-negative. So, the numerator is always non-negative, and the denominator is negative in the interval $0 < x < 36$. Therefore, the entire expression is negative except when the numerator is zero. Therefore, the solution is $0 < x < 36$ excluding $x = 9$.

But let me check if there's a possibility that the absolute value could be zero in the interval $0 < x < 36$. For example, $x = 9$ is in that interval, and that's where the numerator is zero. So, yes, only at $x = 9$. Therefore, the solution is $0 < x < 36$ except $x = 9$.

But let me also 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. Therefore, the final answer is $0 < x < 36$ with $x \neq 9$.

But let me write that in interval notation. So, from 0 to 9, not including 0 and 9, and from 9 to 36, not including 9 and 36. Therefore, $(0, 9) \cup (9, 36)$.

But let me check if there's a mistake here. Suppose someone might think that the absolute value could affect the sign. For example, if the expression inside the absolute value is negative, then the absolute value would make it positive. But since the absolute value is always non-negative, regardless of the inside, the numerator is non-negative. Therefore, the sign of the entire expression is determined by the denominator. Therefore, the only time the expression is negative is when the denominator is negative and the numerator is not zero. Therefore, the solution is as above.

Alternatively, maybe I can approach this problem by considering cases based on the absolute value. Let me try that approach to verify.

The absolute value $|x^2 - 81|$ can be split into two cases:

Case 1: $x^2 - 81 \geq 0$, which implies $x^2 \geq 81$, so $x \geq 9$ or $x \leq -9$.

Case 2: $x^2 - 81 < 0$, which implies $x^2 < 81$, so $-9 < x < 9$.

So, in Case 1: $x^2 - 81 \geq 0$, so $|x^2 - 81| = x^2 - 81$. Therefore, the inequality becomes $\frac{x^2 - 81}{x^2 - 36x} < 0$.

In Case 2: $x^2 - 81 < 0$, so $|x^2 - 81| = -(x^2 - 81) = -x^2 + 81$. Therefore, the inequality becomes $\frac{-x^2 + 81}{x^2 - 36x} < 0$.

So, let me analyze both cases separately.

Starting with Case 1: $x \geq 9$ or $x \leq -9$, and $\frac{x^2 - 81}{x^2 - 36x} < 0$.

First, note that in this case, the numerator is $x^2 - 81$, which is non-negative, and the denominator is $x^2 - 36x$. So, the fraction is negative when the denominator is negative (since numerator is non-negative). Therefore, similar to before, we need $x^2 - 36x < 0$, which is $0 < x < 36$. However, in Case 1, we have $x \geq 9$ or $x \leq -9$. Therefore, the overlap between $x \geq 9$ or $x \leq -9$ and $0 < x < 36$ is $9 \leq x < 36$. However, we also need to remember that in Case 1, the numerator is non-negative, but we need the fraction to be negative, so denominator must be negative. Therefore, the solution in Case 1 is $9 \leq x < 36$. However, we also need to check if the numerator is zero here. The numerator is zero when $x = 9$ or $x = -9$. But in this case, $x \geq 9$ or $x \leq -9$, so $x = 9$ is included here. However, at $x = 9$, the numerator is zero, so the fraction is zero, which is not less than zero. Therefore, we need to exclude $x = 9$. Therefore, in Case 1, the solution is $9 < x < 36$.

Now, moving to Case 2: $-9 < x < 9$, and the inequality is $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me simplify this. The numerator is $-x^2 + 81 = -(x^2 - 81)$. The denominator is $x^2 - 36x$. So, the inequality becomes $\frac{-(x^2 - 81)}{x^2 - 36x} < 0$, which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$ (multiplying numerator and denominator by -1 reverses the inequality sign). Wait, let me check that again.

Original inequality in Case 2: $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me factor numerator and denominator:

Numerator: $-x^2 + 81 = -(x^2 - 81) = -(x - 9)(x + 9)$.

Denominator: $x^2 - 36x = x(x - 36)$.

So, the inequality becomes $\frac{ - (x - 9)(x + 9) }{x(x - 36)} < 0$. Which is equivalent to $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$ (multiplying both sides by -1 reverses the inequality).

So, now we have $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. Let me analyze this inequality. The critical points are where the numerator or denominator is zero: $x = -9$, $x = 0$, $x = 9$, $x = 36$. These divide the real line into intervals:

1. $x < -9$
2. $-9 < x < 0$
3. $0 < x < 9$
4. $9 < x < 36$
5. $x > 36$

But remember, in Case 2, we have $-9 < x < 9$. Therefore, we only need to consider intervals 2, 3, and 4 (but 4 is part of Case 1? Wait, no. Wait, Case 2 is $-9 < x < 9$, so in this case, the intervals to consider are:

- $-9 < x < 0$
- $0 < x < 9$

Because $x = -9$ and $x = 9$ are excluded from Case 2 (since Case 2 is $-9 < x < 9$). Also, $x = 0$ is excluded because denominator is zero there.

So, let me check the sign of $\frac{(x - 9)(x + 9)}{x(x - 36)}$ in these intervals.

First interval: $-9 < x < 0$.

Let me pick $x = -1$:

Numerator: $(-1 - 9)(-1 + 9) = (-10)(8) = -80$

Denominator: $(-1)(-1 - 36) = (-1)(-37) = 37$

So, the fraction is $-80 / 37 < 0$. But we have the inequality $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$. Therefore, this interval does not satisfy the inequality.

Second interval: $0 < x < 9$. Let me pick $x = 1$:

Numerator: $(1 - 9)(1 + 9) = (-8)(10) = -80$

Denominator: $1(1 - 36) = 1(-35) = -35$

So, the fraction is $-80 / -35 = 80/35 > 0$. Therefore, this interval satisfies the inequality.

Third interval: $x = 9$ is excluded. So, in Case 2, we only have $-9 < x < 9$, so the only interval that satisfies the inequality in Case 2 is $0 < x < 9$.

Therefore, combining Case 1 and Case 2:

Case 1 gives $9 < x < 36$.

Case 2 gives $0 < x < 9$.

Therefore, the total solution is $0 < x < 9$ or $9 < x < 36$, which is the same as $(0, 9) \cup (9, 36)$. Therefore, confirming the previous result.

But let me check if there are any other points. For example, in Case 2, when we had $-9 < x < 0$, the inequality was not satisfied. So, nothing there. Also, in Case 1, we had $9 < x < 36$. So, combining both cases, the solution is indeed $(0, 9) \cup (9, 36)$.

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

But let me check if there are any other restrictions. For example, in Case 2, when we had $-9 < x < 9$, but the original inequality is $\frac{|x^2 - 81|}{x^2 - 36x} < 0$. However, in Case 2, we transformed the inequality into $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$, which led us to $0 < x < 9$. But in this interval, the denominator $x^2 - 36x$ is negative (as we saw earlier), and the numerator $|x^2 - 81|$ is positive (since $x^2 - 81 < 0$ in this interval, so absolute value makes it positive). Therefore, the fraction is positive divided by negative, which is negative. Wait, but in Case 2, we had transformed the inequality into $\frac{(x - 9)(x + 9)}{x(x - 36)} > 0$, which is equivalent to the original inequality. Wait, but in Case 2, the original inequality is $\frac{-x^2 + 81}{x^2 - 36x} < 0$, which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$. But in the interval $0 < x < 9$, the numerator $x^2 - 81$ is negative, and the denominator $x^2 - 36x$ is negative. Therefore, negative divided by negative is positive, which satisfies the inequality $\frac{x^2 - 81}{x^2 - 36x} > 0$. Therefore, the original inequality $\frac{-x^2 + 81}{x^2 - 36x} < 0$ is equivalent to this. Therefore, in this interval, the original inequality is satisfied. Therefore, the solution in Case 2 is $0 < x < 9$, and in Case 1, $9 < x < 36$. Therefore, combining both, the solution is $0 < x < 36$ excluding $x = 9$.

Therefore, the answer is $(0, 9) \cup (9, 36)$.

But let me check if there's a mistake in the sign when I transformed the inequality. Let me re-express:

Original inequality in Case 2: $\frac{-x^2 + 81}{x^2 - 36x} < 0$. Let me write this as $\frac{-(x^2 - 81)}{x^2 - 36x} < 0$. Which is equivalent to $\frac{x^2 - 81}{x^2 - 36x} > 0$ because multiplying numerator and denominator by -1 reverses the inequality. Therefore, yes, that's correct. Therefore, solving $\frac{x^2 - 81}{x^2 - 36x} > 0$ in the interval $-9 < x < 9$.

But in this interval, $x^2 - 81 < 0$ and $x^2 - 36x < 0$ (since $0 < x < 36$), so the fraction is negative divided by negative, which is positive. Therefore, the inequality is satisfied. Therefore, the solution in Case 2 is $0 < x < 9$ (since $-9 < x < 0$ gives negative over positive, which is negative, not satisfying the inequality). Therefore, correct.

Therefore, combining both cases, the solution is indeed $0 < x < 9$ and $9 < x < 36$.

Therefore, the final answer is $(0, 9) \cup (9, 36)$.

But let me check if there are any other points where the original expression is defined. For example, at $x = 0$ and $x = 36$, the denominator is zero, so they are excluded. At $x = 9$, the numerator is zero, so the expression is zero, which is not less than zero, so excluded. At $x = -9$, the numerator is zero, but the denominator is positive, so the expression is zero, which is not less than zero, so excluded. Therefore, all other points in the intervals are okay.

Therefore, I think this is correct. Therefore, the solution is $0 < x < 36$ excluding $x = 9$, which is $(0, 9) \cup (9, 36)$.

**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 analyze the behavior of the expression by considering the sign of the numerator and denominator.

---

### Step 1: Understand the Structure

The expression is a rational function with an **absolute value** in the numerator and a **quadratic expression** in the denominator. The absolute value ensures the numerator is **non-negative** (i.e., $ |x^2 - 81| \geq 0 $), and the denominator is a quadratic expression that can be factored as:

$$
x^2 - 36x = x(x - 36).
$$

So, the sign of the entire expression depends on the **sign of the denominator**, since the numerator is always non-negative. The inequality is strict: the expression must be **less than zero**, which only occurs when the **numerator is positive** and the **denominator is negative**.

---

### Step 2: Determine When the Numerator is Zero

The numerator is $ |x^2 - 81| $, which is zero when:

$$
x^2 - 81 = 0 \Rightarrow x = \pm 9.
$$

At these points, the entire expression becomes zero, which **does not satisfy** the inequality $ < 0 $. So, we **exclude** $ x = 9 $ and $ x = -9 $ from the solution set.

---

### Step 3: Determine When the Denominator is Negative

We solve the inequality:

$$
x(x - 36) < 0.
$$

This is a quadratic inequality. The critical points are $ x = 0 $ and $ x = 36 $. Testing intervals:

- $ x < 0 $: $ x $ and $ x - 36 $ are both negative → product is **positive**
- $ 0 < x < 36 $: $ x $ is positive, $ x - 36 $ is negative → product is **negative**
- $ x > 36 $: both are positive → product is **positive**

So, the denominator is **negative** when $ 0 < x < 36 $.

---

### Step 4: Combine Conditions

We now combine the two conditions:

- The **denominator is negative** when $ 0 < x < 36 $
- The **numerator is not zero** when $ x \neq \pm 9 $

So, the expression is **negative** (i.e., satisfies the inequality) when:

$$
0 < x < 36 \quad \text{and} \quad x \neq 9.
$$

This gives the solution set:

$$
(0, 9) \cup (9, 36).
$$

---

### 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, which is less than 160. The difference is 160 - 158.515 = 1.485.

So how much more do we need to add to 4.9 to get the remaining 1.485?

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. Approximately 39.57 USD per day.

Okay, so Navin earns roughly $39.57 per day.

Now moving on to Luka from Croatia, who earns 25 kuna per hour.

Exchange rate: 1 USD = 5.18 kuna.

Therefore, to convert kuna to USD, divide by 5.18.

So 25 / 5.18.

Let me calculate 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.

Alternatively, maybe I can do decimal division.

Let me write 25 divided by 5.18.

Multiply numerator and denominator by 100 to eliminate decimals: 2500 / 518.

Let me compute 518 * 4 = 2072

Subtract from 2500: 2500 - 2072 = 428

Bring down a zero (making it 4280), divide by 518.

518 * 8 = 4144

Subtract: 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 4.826 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. Total is 32 + 6.608 = 38.608 USD. Approximately $38.61 per day.

So Luka earns roughly $38.61 per day.

Now, moving on to Ian from Botswana, who earns 34 pula per hour.

Exchange rate: 1 USD = 6.95 pula.

Therefore, converting pula to USD: 34 / 6.95.

Let me calculate that. Let me do similar steps.

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 ≈ 6.95 * 0.9 = 6.255, which is slightly more than 6.2. So approximately 0.89 - a little.

Let me compute 6.2 / 6.95.

Multiply numerator and denominator by 100: 620 / 695.

Divide numerator and denominator by 5: 124 / 139.

Hmm, not sure. Alternatively, approximate decimal.

Let me note that 6.95 * 0.89 = ?

Well, 6.95 * 0.8 = 5.56

6.95 * 0.09 = 0.6255

Total is 5.56 + 0.6255 = 6.1855

Which is very close to 6.2. So 0.89 gives us 6.1855, which is 6.2 - 6.1855 = 0.0145 less.

So to get the remaining 0.0145, how much more do we need?

Since 6.95 * x = 0.0145, so x ≈ 0.0145 / 6.95 ≈ 0.002086.

Therefore, total multiplier is approximately 0.89 + 0.002086 ≈ 0.8921.

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

So approximately 4.892 USD per hour.

Therefore, Ian's daily earnings would be 4.892 * 8.

Calculating that: 4 * 8 = 32, 0.892 * 8 = 7.136. Total is 32 + 7.136 = 39.136 USD. Approximately $39.14 per day.

So summarizing:

Navin: ~$39.57

Luka: ~$38.61

Ian: ~$39.14

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

Wait, but let me check my calculations again because sometimes when approximating, errors can creep in. Let me verify each calculation with more precise methods or perhaps use fractions.

Starting with Navin again:

Navin: 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 point and add zeros: 30.6 becomes 30.60

Now, 32.35 goes into 306.0 (after multiplying numerator and denominator by 10) how many times?

Wait, perhaps better to do it step by step.

Alternatively, use calculator-like approach.

Let me consider 32.35 × 4.9 = ?

As before, 32.35 × 4 = 129.4, 32.35 × 0.9 = 29.115, total 158.515

Difference between 160 and 158.515 is 1.485.

So 1.485 / 32.35 ≈ 0.0459

So total is 4.9 + 0.0459 ≈ 4.9459, so 4.9459 USD per hour.

Multiply by 8: 4.9459 × 8.

Let me compute 4.9459 × 8:

First, 4 × 8 = 32

0.9459 × 8 = ?

Well, 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 ≈ $39.57. So that's accurate.

Now Luka: 25 / 5.18

Let me compute this division more accurately.

Let me write 25 ÷ 5.18.

Let me convert this to 2500 ÷ 518 (multiplying numerator and denominator by 100).

Let me perform the division 2500 ÷ 518.

First, 518 × 4 = 2072

Subtract from 2500: 2500 - 2072 = 428

Bring down a 0: 4280

518 × 8 = 4144

Subtract: 4280 - 4144 = 136

Bring down a 0: 1360

518 × 2 = 1036

Subtract: 1360 - 1036 = 324

Bring down a 0: 3240

518 × 6 = 3108

Subtract: 3240 - 3108 = 132

Bring down a 0: 1320

518 × 2 = 1036

Subtract: 1320 - 1036 = 284

Bring down a 0: 2840

518 × 5 = 2590

Subtract: 2840 - 2590 = 250

So so far, the division gives us 4.826... So the decimal is approximately 4.8262...

So approximately 4.8262 USD per hour.

Multiply by 8: 4.8262 × 8.

Let me compute that:

4 × 8 = 32

0.8262 × 8 = ?

0.8 × 8 = 6.4

0.0262 × 8 = 0.2096

Total: 6.4 + 0.2096 = 6.6096

Total earnings: 32 + 6.6096 = 38.6096 ≈ $38.61. So that's accurate.

Now Ian: 34 / 6.95

Let me compute this division accurately.

First, 34 ÷ 6.95.

Multiply numerator and denominator by 100: 3400 ÷ 695.

Let me perform this division.

First, 695 × 4 = 2780

Subtract from 3400: 3400 - 2780 = 620

Bring down a 0: 6200

695 × 8 = 5560

Subtract: 6200 - 5560 = 640

Bring down a 0: 6400

695 × 9 = 6255

Subtract: 6400 - 6255 = 145

Bring down a 0: 1450

695 × 2 = 1390

Subtract: 1450 - 1390 = 60

Bring down a 0: 600

695 × 0.8 ≈ 556

So approximately 0.8...

So putting it all together, the division is 4.8 (from 4 + 0.8?), wait let me track:

First step: 4. (since 695 × 4 = 2780)

Then after subtracting, we had 620, then brought down a 0 to make 6200, divided by 695 gives 8, so total so far is 4.8...

Then subtract 5560, remainder 640, bring down 0 to 6400, divided by 695 is 9, so total is 4.89...

Subtract 6255, remainder 145, bring down 0 to 1450, divided by 695 is 2, so total is 4.892...

Subtract 1390, remainder 60, bring down 0 to 600, divided by 695 is 0.8... So approximately 4.892...

So approximately 4.892 USD per hour.

Multiply by 8: 4.892 × 8.

Let me compute that:

4 × 8 = 32

0.892 × 8 = ?

0.8 × 8 = 6.4

0.092 × 8 = 0.736

Total: 6.4 + 0.736 = 7.136

Total earnings: 32 + 7.136 = 39.136 ≈ $39.14.

So that's accurate.

So, Navin: ~$39.57, Luka: ~$38.61, Ian: ~$39.14.

Therefore, Navin earns the most. But let me check once again if there's any miscalculations.

Alternatively, maybe I can use fractions or another method to cross-verify.

Alternatively, let me use exact decimal values with more precision.

Alternatively, perhaps I can use calculator-like computations.

But since I don't have a calculator here, I need to rely on my manual calculations.

Alternatively, let me check the exact value for Ian's hourly wage.

Ian: 34 / 6.95.

Let me compute 34 divided by 6.95.

Let me note that 6.95 is equal to 6 + 0.95.

Alternatively, think of 6.95 as 6.95 = 6 + 0.95 = 6 + 19/20.

But maybe not helpful.

Alternatively, note that 6.95 × 4.89 = ?

Let me compute 6.95 × 4.89.

First, compute 6.95 × 4 = 27.8

6.95 × 0.8 = 5.56

6.95 × 0.09 = 0.6255

Adding them together: 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is very close to 34. So 6.95 × 4.89 ≈ 33.9855, which is just slightly less than 34. The difference is 34 - 33.9855 = 0.0145.

Therefore, to get the exact value, we need to add a little more to 4.89.

So, 0.0145 / 6.95 ≈ 0.002086.

Therefore, total is approximately 4.89 + 0.002086 ≈ 4.892086, which matches our prior calculation of ~4.892 USD per hour. Therefore, Ian's hourly wage is approximately 4.892 USD, so 8 hours would be 39.136 USD, which is ~$39.14.

So that seems correct.

Therefore, Navin's total is ~$39.57, which is higher than Ian's ~$39.14, and Luka's ~$38.61.

Therefore, Navin earns the most.

But let me check once again if I didn't mix up any numbers.

Wait, the exchange rates:

Navin: 1 USD = 32.35 rupee. So 1 rupee = 1/32.35 USD. So 160 rupee is 160 / 32.35 USD. Correct.

Luka: 1 USD = 5.18 kuna. So 25 kuna is 25 / 5.18 USD. Correct.

Ian: 1 USD = 6.95 pula. So 34 pula is 34 / 6.95 USD. Correct.

Yes, that's correct.

So calculations are correct.

Therefore, Navin earns the most.

But let me check if there's any chance that the question is asking for something else, like total earnings in their own currency? But no, the question says "who would earn the most after working an eight-hour day?" and since they are comparing their earnings, it's logical to convert to a common currency (USD) to compare.

Alternatively, maybe the question is in the original currencies? But that would not make sense because the currencies are different. So converting to USD is the right approach.

Alternatively, maybe the question is in terms of their local currency? But then, how would you compare? For example, Navin earns 160 rupee per hour, Luka 25 kuna, Ian 34 pula. But without knowing the exchange rates between their currencies, you can't compare. Therefore, the problem must require converting to USD.

Therefore, the answer is Navin.

But let me check once again the exact numbers with more precise decimal places.

For Navin:

160 / 32.35 = ?

Let me compute 160 / 32.35.

Let me note that 32.35 × 4.94 = ?

Compute 32.35 × 4.94:

First, 32.35 × 4 = 129.4

32.35 × 0.94 = ?

Compute 32.35 × 0.9 = 29.115

32.35 × 0.04 = 1.294

Total: 29.115 + 1.294 = 30.409

Therefore, total 32.35 × 4.94 = 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 approximately 4.94 + 0.0059 ≈ 4.9459 USD per hour, which matches our prior calculation. Therefore, 4.9459 × 8 = 39.5672 USD.

So Navin: ~$39.57

Ian: ~$39.14

So Navin is higher.

Therefore, the answer is Navin.

But let me check if there's any possible mistake in 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 that is:

1 USD = 32.35 rupee => 1 rupee = 1/32.35 USD

Similarly for others.

Yes, that's correct.

Therefore, calculations are correct.

Therefore, Navin earns the most.

But let me check once again with exact fractions.

Alternatively, maybe I can use fractions to compute more accurately.

But that might be too time-consuming. Alternatively, let me check with approximate decimal values.

Alternatively, think of the exact decimal values:

Navin: 160 / 32.35 ≈ 4.9459 USD/hour

Luka: 25 / 5.18 ≈ 4.8262 USD/hour

Ian: 34 / 6.95 ≈ 4.8920 USD/hour

So, 4.9459 > 4.8920 > 4.8262, so Navin > Ian > Luka.

Therefore, after 8 hours:

Navin: ~39.57

Ian: ~39.14

Luka: ~38.61

Therefore, Navin earns the most.

Therefore, the answer is Navin.

But let me check once again if I didn't transpose any numbers.

Original problem:

Navin makes 160 rupee per hour.

Luka makes 25 kuna per hour.

Ian makes 34 pula per hour.

Exchange rates:

1 USD = 32.35 rupee

1 USD = 5.18 kuna

1 USD = 6.95 pula

Yes, that's correct.

So, no, numbers are correct.

Therefore, conclusion: Navin earns the most.

But wait, just to make sure, let me check if the question is asking for the total in their own currency? But that would not make sense. For example, if we compare 160 rupee, 25 kuna, 34 pula, but without knowing the exchange rates between these currencies, we can't compare. Therefore, the only way is to convert to USD.

Therefore, the answer is Navin.

But let me check once again with exact decimal calculations for Ian.

Ian: 34 / 6.95.

Let me compute 34 divided by 6.95.

Let me write 6.95 × 4.89 = ?

As before, 6.95 × 4.89 ≈ 33.9855, which is 34 - 0.0145.

So, 34 / 6.95 = 4.89 + (0.0145 / 6.95) ≈ 4.89 + 0.002086 ≈ 4.892086.

Therefore, 4.892086 × 8 = ?

Let me compute 4.892086 × 8:

First, 4 × 8 = 32

0.892086 × 8:

Compute 0.8 × 8 = 6.4

0.092086 × 8 = 0.736688

Total: 6.4 + 0.736688 = 7.136688

Total earnings: 32 + 7.136688 = 39.136688 ≈ $39.14.

Yes, so that's correct.

Therefore, Navin's total is ~$39.57, which is higher than Ian's ~$39.14.

Therefore, Navin earns the most.

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), and then calculate their total earnings for the day.

---

### **Step 1: Convert Hourly Wages to USD**

We are given the following exchange rates:

- 1 USD = 32.35 Mauritius rupee (Navin's currency)
- 1 USD = 5.18 Croatian kuna (Luka's currency)
- 1 USD = 6.95 Botswana pula (Ian's currency)

We convert each person's hourly wage to USD by dividing their hourly wage by the respective exchange rate.

#### **Navin (Mauritius rupee):**
$$
\text{Hourly wage in USD} = \frac{160}{32.35} \approx 4.946 \text{ USD}
$$
$$
\text{Daily earnings} = 4.946 \times 8 \approx 39.57 \text{ USD}
$$

#### **Luka (Croatian kuna):**
$$
\text{Hourly wage in USD} = \frac{25}{5.18} \approx 4.826 \text{ USD}
$$
$$
\text{Daily earnings} = 4.826 \times 8 \approx 38.61 \text{ USD}
$$

#### **Ian (Botswana pula):**
$$
\text{Hourly wage in USD} = \frac{34}{6.95} \approx 4.892 \text{ USD}
$$
$$
\text{Daily earnings} = 4.892 \times 8 \approx 39.14 \text{ USD}
$$

---

### **Step 2: Compare Daily Earnings**

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

---

### **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, which is less than 160. The difference is 160 - 158.515 = 1.485.

So how much more do we need to add to 4.9 to get the remaining 1.485?

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. Approximately 39.57 USD per day.

Okay, so Navin earns roughly $39.57 per day.

Now moving on to Luka from Croatia, who earns 25 kuna per hour.

Exchange rate: 1 USD = 5.18 kuna.

Therefore, to convert kuna to USD, divide by 5.18.

So 25 / 5.18.

Let me calculate 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.

Alternatively, maybe I can do decimal division.

Let me write 25 divided by 5.18.

Multiply numerator and denominator by 100 to eliminate decimals: 2500 / 518.

Let me compute 518 * 4 = 2072

Subtract from 2500: 2500 - 2072 = 428

Bring down a zero (making it 4280), divide by 518.

518 * 8 = 4144

Subtract: 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 4.826 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. Total is 32 + 6.608 = 38.608 USD. Approximately $38.61 per day.

So Luka earns roughly $38.61 per day.

Now, moving on to Ian from Botswana, who earns 34 pula per hour.

Exchange rate: 1 USD = 6.95 pula.

Therefore, converting pula to USD: 34 / 6.95.

Let me calculate that. Let me do similar steps.

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 ≈ 6.95 * 0.9 = 6.255, which is slightly more than 6.2. So approximately 0.89 - a little.

Let me compute 6.2 / 6.95.

Multiply numerator and denominator by 100: 620 / 695.

Divide numerator and denominator by 5: 124 / 139.

Hmm, not sure. Alternatively, approximate decimal.

Let me note that 6.95 * 0.89 = ?

Well, 6.95 * 0.8 = 5.56

6.95 * 0.09 = 0.6255

Total is 5.56 + 0.6255 = 6.1855

Which is very close to 6.2. So 0.89 gives us 6.1855, which is 6.2 - 6.1855 = 0.0145 less.

So to get the remaining 0.0145, how much more do we need?

Since 6.95 * x = 0.0145, so x ≈ 0.0145 / 6.95 ≈ 0.002086.

Therefore, total multiplier is approximately 0.89 + 0.002086 ≈ 0.8921.

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

So approximately 4.892 USD per hour.

Therefore, Ian's daily earnings would be 4.892 * 8.

Calculating that: 4 * 8 = 32, 0.892 * 8 = 7.136. Total is 32 + 7.136 = 39.136 USD. Approximately $39.14 per day.

So summarizing:

Navin: ~$39.57

Luka: ~$38.61

Ian: ~$39.14

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

Wait, but let me check my calculations again because sometimes when approximating, errors can creep in. Let me verify each calculation with more precise methods or perhaps use fractions.

Starting with Navin again:

Navin: 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 point and add zeros: 30.6 becomes 30.60

Now, 32.35 goes into 306.0 (after multiplying numerator and denominator by 10) how many times?

Wait, perhaps better to do it step by step.

Alternatively, use calculator-like approach.

Let me consider 32.35 × 4.9 = ?

As before, 32.35 × 4 = 129.4, 32.35 × 0.9 = 29.115, total 158.515

Difference between 160 and 158.515 is 1.485.

So 1.485 / 32.35 ≈ 0.0459

So total is 4.9 + 0.0459 ≈ 4.9459, so 4.9459 USD per hour.

Multiply by 8: 4.9459 × 8.

Let me compute 4.9459 × 8:

First, 4 × 8 = 32

0.9459 × 8 = ?

Well, 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 ≈ $39.57. So that's accurate.

Now Luka: 25 / 5.18

Let me compute this division more accurately.

Let me write 25 ÷ 5.18.

Let me convert this to 2500 ÷ 518 (multiplying numerator and denominator by 100).

Let me perform the division 2500 ÷ 518.

First, 518 × 4 = 2072

Subtract from 2500: 2500 - 2072 = 428

Bring down a 0: 4280

518 × 8 = 4144

Subtract: 4280 - 4144 = 136

Bring down a 0: 1360

518 × 2 = 1036

Subtract: 1360 - 1036 = 324

Bring down a 0: 3240

518 × 6 = 3108

Subtract: 3240 - 3108 = 132

Bring down a 0: 1320

518 × 2 = 1036

Subtract: 1320 - 1036 = 284

Bring down a 0: 2840

518 × 5 = 2590

Subtract: 2840 - 2590 = 250

So so far, the division gives us 4.826... So the decimal is approximately 4.8262...

So approximately 4.8262 USD per hour.

Multiply by 8: 4.8262 × 8.

Let me compute that:

4 × 8 = 32

0.8262 × 8 = ?

0.8 × 8 = 6.4

0.0262 × 8 = 0.2096

Total: 6.4 + 0.2096 = 6.6096

Total earnings: 32 + 6.6096 = 38.6096 ≈ $38.61. So that's accurate.

Now Ian: 34 / 6.95

Let me compute this division accurately.

First, 34 ÷ 6.95.

Multiply numerator and denominator by 100: 3400 ÷ 695.

Let me perform this division.

First, 695 × 4 = 2780

Subtract from 3400: 3400 - 2780 = 620

Bring down a 0: 6200

695 × 8 = 5560

Subtract: 6200 - 5560 = 640

Bring down a 0: 6400

695 × 9 = 6255

Subtract: 6400 - 6255 = 145

Bring down a 0: 1450

695 × 2 = 1390

Subtract: 1450 - 1390 = 60

Bring down a 0: 600

695 × 0.8 ≈ 556

So approximately 0.8...

So putting it all together, the division is 4.8 (from 4 + 0.8?), wait let me track:

First step: 4. (since 695 × 4 = 2780)

Then after subtracting, we had 620, then brought down a 0 to make 6200, divided by 695 gives 8, so total so far is 4.8...

Then subtract 5560, remainder 640, bring down 0 to 6400, divided by 695 is 9, so total is 4.89...

Subtract 6255, remainder 145, bring down 0 to 1450, divided by 695 is 2, so total is 4.892...

Subtract 1390, remainder 60, bring down 0 to 600, divided by 695 is 0.8... So approximately 4.892...

So approximately 4.892 USD per hour.

Multiply by 8: 4.892 × 8.

Let me compute that:

4 × 8 = 32

0.892 × 8 = ?

0.8 × 8 = 6.4

0.092 × 8 = 0.736

Total: 6.4 + 0.736 = 7.136

Total earnings: 32 + 7.136 = 39.136 ≈ $39.14.

So that's accurate.

So, Navin: ~$39.57, Luka: ~$38.61, Ian: ~$39.14.

Therefore, Navin earns the most. But let me check once again if there's any miscalculations.

Alternatively, maybe I can use fractions or another method to cross-verify.

Alternatively, let me use exact decimal values with more precision.

Alternatively, perhaps I can use calculator-like computations.

But since I don't have a calculator here, I need to rely on my manual calculations.

Alternatively, let me check the exact value for Ian's hourly wage.

Ian: 34 / 6.95.

Let me compute 34 divided by 6.95.

Let me note that 6.95 is equal to 6 + 0.95.

Alternatively, think of 6.95 as 6.95 = 6 + 0.95 = 6 + 19/20.

But maybe not helpful.

Alternatively, note that 6.95 × 4.89 = ?

Let me compute 6.95 × 4.89.

First, compute 6.95 × 4 = 27.8

6.95 × 0.8 = 5.56

6.95 × 0.09 = 0.6255

Adding them together: 27.8 + 5.56 = 33.36 + 0.6255 = 33.9855

Which is very close to 34. So 6.95 × 4.89 ≈ 33.9855, which is just slightly less than 34. The difference is 34 - 33.9855 = 0.0145.

Therefore, to get the exact value, we need to add a little more to 4.89.

So, 0.0145 / 6.95 ≈ 0.002086.

Therefore, total is approximately 4.89 + 0.002086 ≈ 4.892086, which matches our prior calculation of ~4.892 USD per hour. Therefore, Ian's hourly wage is approximately 4.892 USD, so 8 hours would be 39.136 USD, which is ~$39.14.

So that seems correct.

Therefore, Navin's total is ~$39.57, which is higher than Ian's ~$39.14, and Luka's ~$38.61.

Therefore, Navin earns the most.

But let me check once again if I didn't mix up any numbers.

Wait, the exchange rates:

Navin: 1 USD = 32.35 rupee. So 1 rupee = 1/32.35 USD. So 160 rupee is 160 / 32.35 USD. Correct.

Luka: 1 USD = 5.18 kuna. So 25 kuna is 25 / 5.18 USD. Correct.

Ian: 1 USD = 6.95 pula. So 34 pula is 34 / 6.95 USD. Correct.

Yes, that's correct.

So calculations are correct.

Therefore, Navin earns the most.

But let me check if there's any chance that the question is asking for something else, like total earnings in their own currency? But no, the question says "who would earn the most after working an eight-hour day?" and since they are comparing their earnings, it's logical to convert to a common currency (USD) to compare.

Alternatively, maybe the question is in the original currencies? But that would not make sense because the currencies are different. So converting to USD is the right approach.

Alternatively, maybe the question is in terms of their local currency? But then, how would you compare? For example, Navin earns 160 rupee per hour, Luka 25 kuna, Ian 34 pula. But without knowing the exchange rates between their currencies, you can't compare. Therefore, the problem must require converting to USD.

Therefore, the answer is Navin.

But let me check once again the exact numbers with more precise decimal places.

For Navin:

160 / 32.35 = ?

Let me compute 160 / 32.35.

Let me note that 32.35 × 4.94 = ?

Compute 32.35 × 4.94:

First, 32.35 × 4 = 129.4

32.35 × 0.94 = ?

Compute 32.35 × 0.9 = 29.115

32.35 × 0.04 = 1.294

Total: 29.115 + 1.294 = 30.409

Therefore, total 32.35 × 4.94 = 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 approximately 4.94 + 0.0059 ≈ 4.9459 USD per hour, which matches our prior calculation. Therefore, 4.9459 × 8 = 39.5672 USD.

So Navin: ~$39.57

Ian: ~$39.14

So Navin is higher.

Therefore, the answer is Navin.

But let me check if there's any possible mistake in 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 that is:

1 USD = 32.35 rupee => 1 rupee = 1/32.35 USD

Similarly for others.

Yes, that's correct.

Therefore, calculations are correct.

Therefore, Navin earns the most.

But let me check once again with exact fractions.

Alternatively, maybe I can use fractions to compute more accurately.

But that might be too time-consuming. Alternatively, let me check with approximate decimal values.

Alternatively, think of the exact decimal values:

Navin: 160 / 32.35 ≈ 4.9459 USD/hour

Luka: 25 / 5.18 ≈ 4.8262 USD/hour

Ian: 34 / 6.95 ≈ 4.8920 USD/hour

So, 4.9459 > 4.8920 > 4.8262, so Navin > Ian > Luka.

Therefore, after 8 hours:

Navin: ~39.57

Ian: ~39.14

Luka: ~38.61

Therefore, Navin earns the most.

Therefore, the answer is Navin.

But let me check once again if I didn't transpose any numbers.

Original problem:

Navin makes 160 rupee per hour.

Luka makes 25 kuna per hour.

Ian makes 34 pula per hour.

Exchange rates:

1 USD = 32.35 rupee

1 USD = 5.18 kuna

1 USD = 6.95 pula

Yes, that's correct.

So, no, numbers are correct.

Therefore, conclusion: Navin earns the most.

But wait, just to make sure, let me check if the question is asking for the total in their own currency? But that would not make sense. For example, if we compare 160 rupee, 25 kuna, 34 pula, but without knowing the exchange rates between these currencies, we can't compare. Therefore, the only way is to convert to USD.

Therefore, the answer is Navin.

But let me check once again with exact decimal calculations for Ian.

Ian: 34 / 6.95.

Let me compute 34 divided by 6.95.

Let me write 6.95 × 4.89 = ?

As before, 6.95 × 4.89 ≈ 33.9855, which is 34 - 0.0145.

So, 34 / 6.95 = 4.89 + (0.0145 / 6.95) ≈ 4.89 + 0.002086 ≈ 4.892086.

Therefore, 4.892086 × 8 = ?

Let me compute 4.892086 × 8:

First, 4 × 8 = 32

0.892086 × 8:

Compute 0.8 × 8 = 6.4

0.092086 × 8 = 0.736688

Total: 6.4 + 0.736688 = 7.136688

Total earnings: 32 + 7.136688 = 39.136688 ≈ $39.14.

Yes, so that's correct.

Therefore, Navin's total is ~$39.57, which is higher than Ian's ~$39.14.

Therefore, Navin earns the most.

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), and then calculate their total earnings for the day.

---

### **Step 1: Convert Hourly Wages to USD**

We are given the following exchange rates:

- 1 USD = 32.35 Mauritius rupee (Navin's currency)
- 1 USD = 5.18 Croatian kuna (Luka's currency)
- 1 USD = 6.95 Botswana pula (Ian's currency)

We convert each person's hourly wage to USD by dividing their hourly wage by the respective exchange rate.

#### **Navin (Mauritius rupee):**
$$
\text{Hourly wage in USD} = \frac{160}{32.35} \approx 4.946 \text{ USD}
$$
$$
\text{Daily earnings} = 4.946 \times 8 \approx 39.57 \text{ USD}
$$

#### **Luka (Croatian kuna):**
$$
\text{Hourly wage in USD} = \frac{25}{5.18} \approx 4.826 \text{ USD}
$$
$$
\text{Daily earnings} = 4.826 \times 8 \approx 38.61 \text{ USD}
$$

#### **Ian (Botswana pula):**
$$
\text{Hourly wage in USD} = \frac{34}{6.95} \approx 4.892 \text{ USD}
$$
$$
\text{Daily earnings} = 4.892 \times 8 \approx 39.14 \text{ USD}
$$

---

### **Step 2: Compare Daily Earnings**

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

---

### **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.