If y is an integer, then the least possible value of |23 - 5y| is

1. Translate the problem requirements: We need to find an integer value of y that minimizes the absolute value expression \(|23 - 5\mathrm{y}|\), which means finding the y that makes \(5\mathrm{y}\) closest to 23.
2. Identify the target relationship: Determine what value of y would make \(5\mathrm{y}\) exactly equal to 23, even though this may not be an integer.
3. Find the closest integer options: Since y must be an integer, identify the integer values of y on either side of our target value.
4. Calculate and compare the results: Evaluate \(|23 - 5\mathrm{y}|\) for both integer options to determine which gives the smaller absolute value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's think about what this problem is really asking. We have the expression \(|23 - 5\mathrm{y}|\) and we want to make this as small as possible.

The absolute value \(|23 - 5\mathrm{y}|\) represents the distance between 23 and \(5\mathrm{y}\) on a number line. To minimize this distance, we want \(5\mathrm{y}\) to be as close as possible to 23.

Since y must be an integer, we're looking for the integer value of y that makes \(5\mathrm{y}\) land closest to the target value of 23.

Process Skill: TRANSLATE - Converting the absolute value minimization into the intuitive concept of finding the closest distance

2. Identify the target relationship

If we could choose any value for y (not just integers), what would make \(|23 - 5\mathrm{y}|\) equal to zero?

We'd need: \(23 - 5\mathrm{y} = 0\)
Solving: \(23 = 5\mathrm{y}\)
So: \(\mathrm{y} = 23/5 = 4.6\)

This means if y could be 4.6, then \(5\mathrm{y}\) would equal exactly 23, and our absolute value expression would be zero. But since y must be an integer, we can't use 4.6.

3. Find the closest integer options

Since our ideal value is \(\mathrm{y} = 4.6\), the two closest integers are:

• \(\mathrm{y} = 4\) (which is 0.6 less than our ideal)
• \(\mathrm{y} = 5\) (which is 0.4 more than our ideal)

These are the only two candidates we need to check, since any other integer would be further from 4.6.

Process Skill: CONSIDER ALL CASES - Systematically identifying the relevant integer options

4. Calculate and compare the results

Now let's calculate \(|23 - 5\mathrm{y}|\) for both integer options:

For \(\mathrm{y} = 4\):
\(|23 - 5(4)| = |23 - 20| = |3| = 3\)

For \(\mathrm{y} = 5\):
\(|23 - 5(5)| = |23 - 25| = |-2| = 2\)

Comparing our results:

• When \(\mathrm{y} = 4\), we get \(|23 - 5\mathrm{y}| = 3\)
• When \(\mathrm{y} = 5\), we get \(|23 - 5\mathrm{y}| = 2\)

Since 2 < 3, the minimum value occurs when \(\mathrm{y} = 5\).

Final Answer

The least possible value of \(|23 - 5\mathrm{y}|\) is 2.

This matches answer choice (B) 2.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the optimization goal

Students may think they need to find the value of y that makes the expression inside the absolute value bars equal to zero, without recognizing that y must be an integer. They might solve \(23 - 5\mathrm{y} = 0\) to get \(\mathrm{y} = 4.6\) and incorrectly assume this is the final answer, missing that y must be a whole number.

2. Not connecting absolute value to distance concept

Students may approach this algebraically by trying to remove the absolute value bars through casework (when \(23 - 5\mathrm{y} \geq 0\) vs when \(23 - 5\mathrm{y} < 0\)) instead of recognizing that \(|23 - 5\mathrm{y}|\) represents the distance between 23 and \(5\mathrm{y}\) on a number line. This geometric insight makes finding the minimum much more intuitive.

3. Failing to identify the constraint properly

Some students may overlook that y must be an integer and attempt to minimize the expression using calculus or treating y as a continuous variable, leading them to incorrectly conclude that the minimum value is 0.

Errors while executing the approach

1. Testing too few integer values

Students might only test one integer value (like \(\mathrm{y} = 5\) since it's closest to 4.6) without checking the adjacent integer (\(\mathrm{y} = 4\)) to confirm which gives the smaller result. They need to test both \(\mathrm{y} = 4\) and \(\mathrm{y} = 5\) to compare.

2. Arithmetic errors in computation

When calculating \(|23 - 5\mathrm{y}|\) for the candidate values, students may make simple computational mistakes such as:

• For \(\mathrm{y} = 4\): incorrectly computing \(|23 - 20|\) or getting the multiplication \(5 \times 4\) wrong
• For \(\mathrm{y} = 5\): incorrectly computing \(|23 - 25| = |-2|\) and forgetting that the absolute value of -2 is positive 2

Errors while selecting the answer

1. Selecting the y-value instead of the expression value

After determining that \(\mathrm{y} = 5\) gives the minimum, students might mistakenly select 5 as their final answer instead of the actual minimum value of the expression \(|23 - 5\mathrm{y}| = 2\). They confuse the input value that produces the minimum with the minimum value itself.