If x and (y - x > 5, where x is an even integer and y and z are odd...

1. Translate the problem requirements: We need to find the smallest possible value of \(\mathrm{z - x}\) given that: \(\mathrm{x < y < z}\) (all in increasing order), \(\mathrm{y - x > 5}\) (difference between y and x exceeds 5), x is even, and both y and z are odd integers.
2. Establish the minimum gap between x and y: Since \(\mathrm{y - x > 5}\) and both x (even) and y (odd) are integers, determine the smallest possible value for \(\mathrm{y - x}\).
3. Find the minimum gap between y and z: Since \(\mathrm{y < z}\) and both are odd integers, determine the smallest possible difference \(\mathrm{z - y}\).
4. Calculate the minimum total gap: Combine the minimum gaps to find the least possible value of \(\mathrm{z - x}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to find step by step. We want the smallest possible value of \(\mathrm{z - x}\), which is the difference between our largest and smallest numbers.

Here are our constraints in plain English:

• We have three numbers x, y, and z where x is the smallest, y is in the middle, and z is the largest
• The difference between y and x must be more than 5
• x must be an even integer (like 2, 4, 6, 8, etc.)
• Both y and z must be odd integers (like 1, 3, 5, 7, etc.)

Mathematically: \(\mathrm{x < y < z, y - x > 5}\), x is even, y and z are odd

Process Skill: TRANSLATE - Converting the word problem into clear mathematical constraints

2. Establish the minimum gap between x and y

Now let's think about the smallest possible difference between y and x. We know that \(\mathrm{y - x > 5}\), which means \(\mathrm{y - x}\) must be at least 6.

But wait - there's something important to consider here. Since x is even and y is odd, let's think about what happens when we subtract an even number from an odd number. The result is always odd!

For example:

• If \(\mathrm{x = 2}\) (even) and \(\mathrm{y = 9}\) (odd), then \(\mathrm{y - x = 7}\) (odd)
• If \(\mathrm{x = 4}\) (even) and \(\mathrm{y = 11}\) (odd), then \(\mathrm{y - x = 7}\) (odd)

Since \(\mathrm{y - x}\) must be greater than 5 AND must be odd (because odd minus even equals odd), the smallest possible value for \(\mathrm{y - x}\) is 7.

Let's verify with an example: If \(\mathrm{x = 2}\), then y must be at least \(\mathrm{2 + 7 = 9}\).

Process Skill: INFER - Recognizing that the parity (even/odd nature) of numbers creates additional constraints

3. Find the minimum gap between y and z

Now let's look at the gap between y and z. We know that \(\mathrm{y < z}\) and both are odd integers.

Since both y and z are odd, what's the smallest possible difference? Well, if we start with any odd number, the very next odd number is 2 more. For example:

• If \(\mathrm{y = 9}\), then the next odd number is \(\mathrm{z = 11}\), giving us \(\mathrm{z - y = 2}\)
• If \(\mathrm{y = 15}\), then the next odd number is \(\mathrm{z = 17}\), giving us \(\mathrm{z - y = 2}\)

So the minimum value for \(\mathrm{z - y}\) is 2.

4. Calculate the minimum total gap

Now we can put it all together! We want to find the minimum value of \(\mathrm{z - x}\).

We can think of this as: \(\mathrm{z - x = (z - y) + (y - x)}\)

From our work above:

• The minimum value of \(\mathrm{(y - x)}\) is 7
• The minimum value of \(\mathrm{(z - y)}\) is 2

Therefore, the minimum value of \(\mathrm{z - x = 7 + 2 = 9}\)

Let's verify with a concrete example:

• Let \(\mathrm{x = 2}\) (even)
• Let \(\mathrm{y = 9}\) (odd, and \(\mathrm{9 - 2 = 7 > 5}\)) ✓
• Let \(\mathrm{z = 11}\) (odd, and \(\mathrm{11 > 9}\)) ✓
• Then \(\mathrm{z - x = 11 - 2 = 9}\)

Process Skill: APPLY CONSTRAINTS - Systematically combining all constraints to find the minimum possible value

4. Final Answer

The least possible value of \(\mathrm{z - x}\) is 9.

Looking at our answer choices:

1. 6
2. 7
3. 8
4. 9
5. 10

The answer is D. 9.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint y - x > 5

Students often interpret "\(\mathrm{y - x > 5}\)" as "\(\mathrm{y - x ≥ 5}\)", thinking the minimum difference can be exactly 5. This leads them to incorrectly assume that \(\mathrm{y - x}\) can equal 5, when it must actually be greater than 5.

2. Overlooking the parity constraint between even and odd integers

Many students fail to recognize that when subtracting an even integer from an odd integer (\(\mathrm{y - x}\)), the result must always be odd. They might think \(\mathrm{y - x}\) could be 6 (the smallest integer greater than 5) without realizing that 6 is even and therefore impossible given the constraints.

3. Attempting to minimize each variable individually rather than the target expression

Students sometimes try to find the smallest possible values for x, y, and z separately, rather than focusing on minimizing the specific expression \(\mathrm{z - x}\). This can lead to suboptimal combinations that don't actually minimize the target difference.

Errors while executing the approach

1. Incorrect calculation of minimum odd difference

When determining that \(\mathrm{y - x}\) must be odd and greater than 5, students might incorrectly conclude the minimum is 5 or 6, forgetting that the next odd number after 5 is 7. They may also make arithmetic errors when adding constraints.

2. Wrong gap calculation between consecutive odd integers

Students sometimes think consecutive odd integers differ by 1 (like consecutive integers) rather than 2. For example, they might think if \(\mathrm{y = 9}\), then z could be 10, forgetting that z must also be odd.

Errors while selecting the answer

1. Selecting a partial result instead of the final answer

Students might select 7 (the minimum value of \(\mathrm{y - x}\)) or 2 (the minimum value of \(\mathrm{z - y}\)) instead of their sum, which gives the minimum value of \(\mathrm{z - x = 9}\). They lose track of what the question is actually asking for.

Alternate Solutions

Smart Numbers Approach

Instead of working with general variables, we can find the minimum value of \(\mathrm{z - x}\) by choosing the smallest possible concrete values that satisfy all constraints.

Step 1: Choose the smallest convenient value for x
Since x must be even, let's use \(\mathrm{x = 0}\) (the smallest non-negative even integer for simplicity).

Step 2: Find the smallest possible y
We need \(\mathrm{y - x > 5}\), so \(\mathrm{y - 0 > 5}\), which means \(\mathrm{y > 5}\).
Since y must be odd and \(\mathrm{y > 5}\), the smallest possible value is \(\mathrm{y = 7}\).
Check: \(\mathrm{y - x = 7 - 0 = 7 > 5}\) ✓

Step 3: Find the smallest possible z
We need \(\mathrm{z > y}\) and z must be odd.
Since \(\mathrm{y = 7}\) and \(\mathrm{z > 7}\) with z odd, the smallest possible value is \(\mathrm{z = 9}\).

Step 4: Calculate z - x
\(\mathrm{z - x = 9 - 0 = 9}\)

Verification with different starting point:
Let's verify with \(\mathrm{x = 2}\):

• y must satisfy \(\mathrm{y - 2 > 5}\), so \(\mathrm{y > 7}\). Since y is odd, minimum \(\mathrm{y = 9}\)
• z must satisfy \(\mathrm{z > 9}\) and be odd, so minimum \(\mathrm{z = 11}\)
• Therefore \(\mathrm{z - x = 11 - 2 = 9}\)

The pattern holds: regardless of which even integer we choose for x, the minimum value of \(\mathrm{z - x}\) is always 9.