sqrt(x^2-1) - sqrt(x^3) +sqrt(x-1) <0

solve this not grahiccaly

We are tasked with solving the inequality:

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

To solve this algebraically, let's proceed step by step:

### Step 1: Identify the domains
Each square root function must have non-negative values inside. This gives us constraints on \( x \):

- For \( \sqrt{x^2 - 1} \), we need \( x^2 - 1 \geq 0 \), so \( x \geq 1 \) or \( x \leq -1 \).
- For \( \sqrt{x^3} \), we need \( x^3 \geq 0 \), so \( x \geq 0 \).
- For \( \sqrt{x - 1} \), we need \( x - 1 \geq 0 \), so \( x \geq 1 \).

### Step 2: Intersection of domains
From the above conditions, we find that the domain of the inequality is \( x \geq 1 \).

### Step 3: Simplify the inequality
Within the domain \( x \geq 1 \), we now analyze the inequality:

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

Since \( x \geq 1 \), each square root term is defined. Let’s attempt substitution for \( x = 1 \) first:

#### At \( x = 1 \):
\[
\sqrt{1^2 - 1} - \sqrt{1^3} + \sqrt{1 - 1} = \sqrt{0} - \sqrt{1} + \sqrt{0} = 0 - 1 + 0 = -1
\]

This satisfies the inequality because \(-1 < 0\).

### Step 4: Test values greater than 1
Now, let's examine what happens when \( x > 1 \). For simplicity, let's try a specific value, say \( x = 2 \):

\[
\sqrt{2^2 - 1} - \sqrt{2^3} + \sqrt{2 - 1} = \sqrt{4 - 1} - \sqrt{8} + \sqrt{1}
\]
\[
= \sqrt{3} - 2\sqrt{2} + 1 \approx 1.732 - 2.828 + 1 = -0.096
\]

This value is still negative, so the inequality holds for \( x = 2 \).

### Step 5: General conclusion
Given that the inequality holds for \( x = 1 \) and for values greater than 1 (as tested with \( x = 2 \)), we conclude that the inequality holds for all \( x \geq 1 \).

### Final Solution:
The solution to the inequality is:
\[
x \geq 1
\]

Would you like more details on any step, or do you have any further questions?

### Related questions:
1. What are the properties of square root functions and their domains?
2. How do you determine the domain of an equation involving radicals?
3. How can inequalities involving radicals be solved graphically?
4. What are the limitations of solving inequalities without graphs?
5. Can inequalities involving cube roots behave differently than square roots?

### Tip:
Always start solving inequalities involving square roots by finding the domain first—it can greatly simplify the problem.

sqrt(x^2-1) - sqrt(x^3) +sqrt(x-1) <0 solve this not graphically

Explore how to solve the inequality sqrt(x^2 - 1) - sqrt(x^3) + sqrt(x - 1) < 0 algebraically. This problem involves key algebraic concepts such as radical functions, domain restrictions, and inequality solving. Suitable for high school students (Grades 10-12).

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