If y geq 0, what is the value of x? |x - 3| geq y |x - 3| leq -y...

Understanding the Question

We need to find the exact value of x, given that \(\mathrm{y} \geq 0\).

This is a value question - we need to determine one specific value for x to have sufficiency.

Given Information

• \(\mathrm{y} \geq 0\) (y is non-negative)
• We need a unique value for x

What We Need to Determine

For sufficiency, we need information that either:

• Directly tells us what x equals, or
• Creates constraints that lead to exactly one possible value of x

Analyzing Statement 1

Statement 1: \(|\mathrm{x} - 3| \geq \mathrm{y}\)

What Statement 1 Tells Us

This tells us that the distance between x and 3 is at least y. Since we know \(\mathrm{y} \geq 0\), this means x is at least y units away from 3.

Testing Different Scenarios

Let's test with different values of y:

• If \(\mathrm{y} = 0\): Then \(|\mathrm{x} - 3| \geq 0\), which is true for ALL values of x (since absolute value is always non-negative)
• If \(\mathrm{y} = 2\): Then \(|\mathrm{x} - 3| \geq 2\), meaning \(\mathrm{x} \geq 5\) or \(\mathrm{x} \leq 1\)

In both cases, we get infinitely many possible values for x.

Conclusion

Statement 1 is NOT sufficient because x can take many different values.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2: \(|\mathrm{x} - 3| \leq -\mathrm{y}\)

What Statement 2 Provides

This might seem tricky at first, but here's where it gets interesting. We know that:

• \(\mathrm{y} \geq 0\), so \(-\mathrm{y} \leq 0\) (non-positive)
• \(|\mathrm{x} - 3| \geq 0\) always (absolute values are never negative)

The Key Insight

We need \(|\mathrm{x} - 3| \leq -\mathrm{y}\), where:

• The left side (\(|\mathrm{x} - 3|\)) is always \(\geq 0\)
• The right side (\(-\mathrm{y}\)) is always \(\leq 0\)

Critical observation: The only way a non-negative value can be less than or equal to a non-positive value is if both equal zero!

This forces:

• \(|\mathrm{x} - 3| = 0\), which gives us \(\mathrm{x} = 3\)
• \(-\mathrm{y} = 0\), which gives us \(\mathrm{y} = 0\)

Let's verify: When \(\mathrm{x} = 3\) and \(\mathrm{y} = 0\):

• \(|3 - 3| = 0\)
• \(-0 = 0\)
• Indeed, \(0 \leq 0\) ✓

Conclusion

Statement 2 is sufficient because it forces \(\mathrm{x} = 3\). [STOP - Sufficient!]

This eliminates choices C and E.

The Answer: B

Statement 2 alone gives us the unique value \(\mathrm{x} = 3\), while Statement 1 alone allows infinitely many values.

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