On the number line, the distance between x and y is greater than the distance between x and z. Does...

Understanding the Question

We need to determine whether z lies between x and y on the number line.

Given information:

• The distance between x and y is greater than the distance between x and z
• In mathematical terms: \(|\mathrm{x} - \mathrm{y}| > |\mathrm{x} - \mathrm{z}|\)

What "between" means:
For z to lie between x and y, one of these must be true:

• \(\mathrm{x} < \mathrm{z} < \mathrm{y}\) (if x is to the left of y), OR
• \(\mathrm{y} < \mathrm{z} < \mathrm{x}\) (if y is to the left of x)

This is a yes/no question. To be sufficient, we need to definitively answer either "yes, z is always between x and y" or "no, z is not always between x and y."

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{xyz} < 0\)

This means the product of all three numbers is negative. For a product of three numbers to be negative, either:

• Exactly one number is negative (and two are positive), OR
• All three numbers are negative

Let's test different scenarios to see if z must lie between x and y:

Test Case 1: \(\mathrm{x} = -2\), \(\mathrm{y} = 3\), \(\mathrm{z} = 1\)

• Check xyz: \((-2)(3)(1) = -6 < 0\) ✓
• Check given constraint: \(|\mathrm{x} - \mathrm{y}| = |-2 - 3| = 5\) and \(|\mathrm{x} - \mathrm{z}| = |-2 - 1| = 3\)
• Since \(5 > 3\), the constraint is satisfied ✓
• Position on number line: x(-2) ... z(1) ... y(3)
• Does z lie between x and y? YES

Test Case 2: \(\mathrm{x} = 4\), \(\mathrm{y} = -1\), \(\mathrm{z} = 2\)

• Check xyz: \((4)(-1)(2) = -8 < 0\) ✓
• Check given constraint: \(|\mathrm{x} - \mathrm{y}| = |4 - (-1)| = 5\) and \(|\mathrm{x} - \mathrm{z}| = |4 - 2| = 2\)
• Since \(5 > 2\), the constraint is satisfied ✓
• Position on number line: y(-1) ... z(2) ... x(4)
• Does z lie between x and y? NO (z is outside the interval from y to x)

Since we found both YES and NO answers, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(\mathrm{xy} < 0\)

This means x and y have opposite signs - one is positive and the other is negative.

Let's test whether this forces z to be between x and y:

Test Case 1: \(\mathrm{x} = -2\), \(\mathrm{y} = 3\) (opposite signs ✓)

• Let \(\mathrm{z} = 1\)
• Check constraint: \(|\mathrm{x} - \mathrm{y}| = 5\) and \(|\mathrm{x} - \mathrm{z}| = 3\), so \(5 > 3\) ✓
• Position: x(-2) ... z(1) ... y(3)
• Does z lie between x and y? YES

Test Case 2: \(\mathrm{x} = -2\), \(\mathrm{y} = 3\) (same x and y values)

• Let \(\mathrm{z} = -3\)
• Check constraint: \(|\mathrm{x} - \mathrm{y}| = 5\) and \(|\mathrm{x} - \mathrm{z}| = |-2 - (-3)| = 1\), so \(5 > 1\) ✓
• Position: z(-3) ... x(-2) ... y(3)
• Does z lie between x and y? NO (z is to the left of both x and y)

Since we found both YES and NO answers, Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now we use both statements together:

• \(\mathrm{xyz} < 0\) (Statement 1)
• \(\mathrm{xy} < 0\) (Statement 2)

Key deduction: Since \(\mathrm{xy} < 0\) (negative), and we need \(\mathrm{xyz} < 0\) (also negative), we must have:

• \(\mathrm{z} > 0\) (positive)

This is because: (negative) × (positive) = (negative)

So our constraints are:

• x and y have opposite signs (one positive, one negative)
• z must be positive
• \(|\mathrm{x} - \mathrm{y}| > |\mathrm{x} - \mathrm{z}|\)

Let's test if this determines z's position:

Test Case 1: \(\mathrm{x} = -2\), \(\mathrm{y} = 3\), \(\mathrm{z} = 1\)

• \(\mathrm{xy} = -6 < 0\) ✓
• \(\mathrm{xyz} = -6 < 0\) ✓
• \(|\mathrm{x} - \mathrm{y}| = 5 > |\mathrm{x} - \mathrm{z}| = 3\) ✓
• Position: x(-2) ... z(1) ... y(3)
• z lies between x and y? YES

Test Case 2: \(\mathrm{x} = 3\), \(\mathrm{y} = -2\), \(\mathrm{z} = 0.5\)

• \(\mathrm{xy} = -6 < 0\) ✓
• \(\mathrm{xyz} = -3 < 0\) ✓
• \(|\mathrm{x} - \mathrm{y}| = 5 > |\mathrm{x} - \mathrm{z}| = 2.5\) ✓
• Position: y(-2) ... z(0.5) ... x(3)
• z lies between x and y? NO (when we swap x and y, z can still be outside the interval)

Even with both statements combined, we can still get both YES and NO answers. The statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

Answer Choice E: "The statements together are not sufficient."