Is \(\mathrm{x} ? \(\mathrm{x}^3(1 - \mathrm{x}^2) \(\mathrm{x}^2 - 1...

Understanding the Question

We need to determine if x is negative.

This is a yes/no question. For a statement to be sufficient, it must allow us to definitively answer either "yes, x is negative" or "no, x is not negative."

Since we have no constraints on x from the question itself, we'll need to rely entirely on what the statements tell us.

Analyzing Statement 1

What Statement 1 Tells Us: \(\mathrm{x}^3(1 - \mathrm{x}^2) < 0\)

Let's think about when this product is negative. For a product to be negative, we need the factors to have opposite signs.

Here's the key insight: \(\mathrm{x}^3\) has the same sign as x (since odd powers preserve sign), while \((1 - \mathrm{x}^2)\) is positive when \(\mathrm{x}^2 < 1\) (that is, when \(-1 < \mathrm{x} < 1\)) and negative when \(\mathrm{x}^2 > 1\) (that is, when \(\mathrm{x} < -1\) or \(\mathrm{x} > 1\)).

Testing Different Scenarios:

Let's test a negative value where \(|\mathrm{x}| < 1\):

• If \(\mathrm{x} = -0.5\):
• \(\mathrm{x}^3 = -0.125\) (negative, same sign as x)
• \((1 - \mathrm{x}^2) = 1 - 0.25 = 0.75\) (positive)
• Product: (negative) × (positive) = negative ✓

Now let's test a positive value where \(|\mathrm{x}| > 1\):

• If \(\mathrm{x} = 2\):
• \(\mathrm{x}^3 = 8\) (positive, same sign as x)
• \((1 - \mathrm{x}^2) = 1 - 4 = -3\) (negative)
• Product: (positive) × (negative) = negative ✓

Since both a negative x (like -0.5) and a positive x (like 2) can make the expression negative, we get different answers to our question "Is \(\mathrm{x} < 0\)?"

• With \(\mathrm{x} = -0.5\): Yes, \(\mathrm{x} < 0\)
• With \(\mathrm{x} = 2\): No, \(\mathrm{x} > 0\)

[STOP - Not Sufficient!]

Conclusion: Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

What Statement 2 Tells Us: \(\mathrm{x}^2 - 1 < 0\)

Rearranging: \(\mathrm{x}^2 < 1\)

This tells us that \(|\mathrm{x}| < 1\), which means \(-1 < \mathrm{x} < 1\).

Testing Different Scenarios:

Within this range, x could be:

• Negative: \(\mathrm{x} = -0.5\) satisfies \(\mathrm{x}^2 - 1 < 0\) (since \(0.25 - 1 = -0.75 < 0\)), and \(\mathrm{x} < 0\) ✓
• Positive: \(\mathrm{x} = 0.5\) satisfies \(\mathrm{x}^2 - 1 < 0\) (since \(0.25 - 1 = -0.75 < 0\)), but \(\mathrm{x} > 0\) ✗

Since x can be either negative or positive while satisfying Statement 2, we cannot definitively answer whether \(\mathrm{x} < 0\).

[STOP - Not Sufficient!]

Conclusion: Statement 2 is NOT sufficient.

This eliminates choice B (and confirms D is already eliminated).

Combining Statements

Now let's see what happens when we use both statements together.

From Statement 2, we know: \(-1 < \mathrm{x} < 1\) (x is between -1 and 1)

From our analysis of Statement 1, we found that \(\mathrm{x}^3(1 - \mathrm{x}^2) < 0\) when:

• x is negative AND \(|\mathrm{x}| < 1\) (like \(\mathrm{x} = -0.5\)), OR
• x is positive AND \(|\mathrm{x}| > 1\) (like \(\mathrm{x} = 2\))

But here's the crucial insight: Statement 2 restricts us to \(|\mathrm{x}| < 1\). This means the second possibility (positive x with \(|\mathrm{x}| > 1\)) is impossible under Statement 2's constraint!

Therefore, the only way to satisfy both statements is if x is negative with \(|\mathrm{x}| < 1\). In other words, \(-1 < \mathrm{x} < 0\).

[STOP - Sufficient!]

Conclusion: When we combine both statements, x must be negative. The statements together are sufficient.

This eliminates choice E.

The Answer: C

Both statements together tell us definitively that x is negative, but neither statement alone provides this certainty.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."