Can you solve this math problem?  sqrt(x^2-1) - sqrt(x^3) +sqrt(x-1) <0

Let's solve the inequality:

\[
\sqrt{x^2 - 1} - \sqrt{x^3} + \sqrt{x - 1} < 0
\]

### Step 1: Analyze the domains of the square roots

For the inequality to be defined:

1. \(\sqrt{x^2 - 1}\) requires \(x^2 - 1 \geq 0\), which simplifies to \(x \leq -1\) or \(x \geq 1\).
2. \(\sqrt{x^3}\) requires \(x^3 \geq 0\), which means \(x \geq 0\).
3. \(\sqrt{x - 1}\) requires \(x - 1 \geq 0\), which means \(x \geq 1\).

The intersection of these domains is \(x \geq 1\).

### Step 2: Simplify the inequality for \(x \geq 1\)

Given that \(x \geq 1\), the inequality becomes:

\[
\sqrt{x^2 - 1} - \sqrt{x^3} + \sqrt{x - 1} < 0
\]

### Step 3: Consider specific values of \(x\) and simplify further

Let's examine the behavior of the expression by substituting values of \(x \geq 1\).

#### When \(x = 1\):
\[
\sqrt{1^2 - 1} - \sqrt{1^3} + \sqrt{1 - 1} = \sqrt{0} - \sqrt{1} + \sqrt{0} = 0 - 1 + 0 = -1
\]
This is less than 0, so the inequality holds for \(x = 1\).

#### When \(x > 1\):
For \(x > 1\), let's analyze the expression's growth:

1. \(\sqrt{x^2 - 1}\) increases as \(x\) increases.
2. \(\sqrt{x^3}\) also increases, but it increases faster than \(\sqrt{x^2 - 1}\) since \(x^3\) grows faster than \(x^2\).
3. \(\sqrt{x - 1}\) increases, but at a slower rate compared to the other two terms.

Since \(\sqrt{x^3}\) grows faster than \(\sqrt{x^2 - 1}\) and \(\sqrt{x - 1}\), the entire expression is expected to become positive as \(x\) increases.

### Step 4: Conclusion

The inequality \(\sqrt{x^2 - 1} - \sqrt{x^3} + \sqrt{x - 1} < 0\) is only satisfied at \(x = 1\). For any \(x > 1\), the expression becomes positive.

Thus, the solution is:

\[
\boxed{x = 1}
\]

Would you like more details or have any other questions?

Here are 5 questions related to this problem:

1. What are the conditions for an expression involving square roots to be defined?
2. How does the growth rate of functions affect the sign of inequalities?
3. How can we graphically represent the inequality \(\sqrt{x^2 - 1} - \sqrt{x^3} + \sqrt{x - 1}\)?
4. What is the significance of the domain in solving inequalities involving radicals?
5. How does the inequality change if we replace the square roots with different roots (e.g., cube roots)?

**Tip:** Always check the domain of the variables before solving inequalities, especially when they involve square roots or other functions with restricted domains.

Can you solve this math problem? sqrt(x^2-1) - sqrt(x^3) +sqrt(x-1) <0

Discover how to solve the inequality sqrt(x^2 - 1) - sqrt(x^3) + sqrt(x - 1) < 0 with a detailed breakdown. This problem involves key algebraic concepts such as radical expressions, inequalities, and the domain of functions. Appropriate for Grades 10-12.

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