If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and...

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Understanding the Question

Let's clarify what we're being asked:

x is a positive number less than 10
We need to determine if \(\mathrm{z} > \frac{\mathrm{x} + 10}{2}\)


In simpler terms: Is z greater than the midpoint between x and 10?

Since \(\mathrm{x} < 10\), this midpoint lies somewhere between x and 10 on the number line. The question asks whether z is to the right of this midpoint.

For this yes/no question to be sufficient, we need to answer either definitively YES or definitively NO.

Analyzing Statement 1

Statement 1 tells us: z is closer to 10 than it is to x.

What This Means

Picture a number line with x on the left and 10 on the right. The midpoint between them is the exact spot where the distances to x and 10 are equal. If z is closer to 10 than to x, where must z be?

The key insight: z must have passed the midpoint! Why? Because:

At the midpoint, distances to x and 10 are equal
To be closer to 10, z must be beyond this balance point, toward 10


Conclusion

Since z is past the midpoint between x and 10, we know \(\mathrm{z} > \frac{\mathrm{x} + 10}{2}\).

This gives us a definitive YES.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

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

Statement 2 tells us: \(\mathrm{z} = 5\mathrm{x}\)

The Key Question

We know z is always 5 times x, but whether this puts z past the midpoint depends on x's specific value. Let's think about this proportionally.

Testing Different Scenarios

Let's test extreme cases to see what happens:

When \(\mathrm{x} = 1\):

\(\mathrm{z} = 5(1) = 5\)
Midpoint = \(\frac{1 + 10}{2} = 5.5\)
Is \(5 > 5.5\)? NO


When \(\mathrm{x} = 2\):

\(\mathrm{z} = 5(2) = 10\)
Midpoint = \(\frac{2 + 10}{2} = 6\)
Is \(10 > 6\)? YES


We get different answers depending on x's value!

Why This Happens

When x is small (like 1), multiplying by 5 doesn't push z far enough to pass the midpoint. But when x is larger (like 2), multiplying by 5 pushes z well past the midpoint.

Conclusion

Since we can get both YES and NO answers depending on x's value, Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Since Statement 1 alone is sufficient but Statement 2 alone is not sufficient, the answer is A.

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