If x is an integer, what is the value of x? x^2 - 4x + 3 (x^2 + 4x +...

Understanding the Question

We need to find the exact value of the integer x.

Given information:

• \(\mathrm{x}\) must be an integer (crucial constraint!)

For sufficiency, we need information that narrows down \(\mathrm{x}\) to exactly one value. If multiple integer values are possible, that's NOT sufficient.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{x}^2 - 4\mathrm{x} + 3 < 0\)

Let me test small integers to see which ones make this expression negative:

• When \(\mathrm{x} = 0\): \(0 - 0 + 3 = 3\) (positive) ✗
• When \(\mathrm{x} = 1\): \(1 - 4 + 3 = 0\) (zero, not negative) ✗
• When \(\mathrm{x} = 2\): \(4 - 8 + 3 = -1\) (negative) ✓
• When \(\mathrm{x} = 3\): \(9 - 12 + 3 = 0\) (zero, not negative) ✗
• When \(\mathrm{x} = 4\): \(16 - 16 + 3 = 3\) (positive) ✗

The key insight: Only \(\mathrm{x} = 2\) makes the expression negative. We've found exactly one integer value that satisfies the condition.

Statement 1 is SUFFICIENT - it gives us a unique value for \(\mathrm{x}\).

[STOP - Sufficient!] This eliminates choices B, C, and E. We only need to check Statement 2 to decide between A and D.

Analyzing Statement 2

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

Statement 2 tells us: \(\mathrm{x}^2 + 4\mathrm{x} + 3 > 0\)

Again, let me test strategic integer values:

• When \(\mathrm{x} = -4\): \(16 - 16 + 3 = 3\) (positive) ✓
• When \(\mathrm{x} = -3\): \(9 - 12 + 3 = 0\) (zero, not positive) ✗
• When \(\mathrm{x} = -2\): \(4 - 8 + 3 = -1\) (negative) ✗
• When \(\mathrm{x} = -1\): \(1 - 4 + 3 = 0\) (zero, not positive) ✗
• When \(\mathrm{x} = 0\): \(0 + 0 + 3 = 3\) (positive) ✓
• When \(\mathrm{x} = 1\): \(1 + 4 + 3 = 8\) (positive) ✓
• When \(\mathrm{x} = 2\): \(4 + 8 + 3 = 15\) (positive) ✓

Notice the pattern: The expression is positive for all integers except -3, -2, and -1. This means \(\mathrm{x}\) could be ..., -5, -4, 0, 1, 2, 3, 4, ... — infinitely many possibilities!

Statement 2 is NOT SUFFICIENT - it allows multiple values for \(\mathrm{x}\).

[STOP - Not sufficient!] This eliminates choices B and D.

The Answer: A

Statement 1 alone gives us exactly one value (\(\mathrm{x} = 2\)), while Statement 2 alone allows multiple values.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."