Question

related questions: lesson 2: expressions

select the answer from the drop down menu.

q. choose different values for \\(x\\) and \\(y\\) from the integers \\(-5, -3, -1, 0, 1, 3,\\) and \\(5\\). what is the greatest value the expression below can have?

\\(-|-2x + y|\\)

Explanation:

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<reasoning>

Analyze the target expression

Using the Integer Operations knowledge point
The expression is:
\[
E = -|-2x + y|
\]
We want to find the greatest possible value of \(E\). Since the absolute value \(|-2x + y|\) is always non-negative (\(\ge 0\)), its negative \(-|-2x + y|\) must be non-positive (\(\le 0\)).
Thus, the maximum possible value of \(E\) is \(0\), which occurs if and only if:
\[
-2x + y = 0 \implies y = 2x
\]

Check for a zero value

We need to find distinct integers \(x\) and \(y\) from the set \(S = \{-5, -3, -1, 0, 1, 3, 5\}\) such that \(y = 2x\).
Let's test possible values for \(x \in S\):

• If \(x = -5\), then \(y = -10

otin S\).

• If \(x = -3\), then \(y = -6

otin S\).

• If \(x = -1\), then \(y = -2

otin S\).

• If \(x = 0\), then \(y = 0\). But \(x\) and \(y\) must be different values, so \(x

e y\). Thus, this pair is invalid.

• If \(x = 1\), then \(y = 2

otin S\).

• If \(x = 3\), then \(y = 6

otin S\).

• If \(x = 5\), then \(y = 10

otin S\).

Since no distinct pair satisfies \(y = 2x\), the expression \(E\) cannot equal \(0\).

Minimize the absolute value

To make \(E = -|-2x + y|\) as large as possible, we must make the absolute value \(|-2x + y|\) as small as possible. Since \(|-2x + y|\) cannot be \(0\), we look for distinct \(x, y \in S\) that make \(|-2x + y|\) equal to \(1\), \(2\), or the smallest possible positive integer.

Let's test if we can get \(|-2x + y| = 1\):
This means \(-2x + y = 1\) or \(-2x + y = -1\).

• Case 1: \(y = 2x + 1\)
• If \(x = -1\), then \(y = 2(-1) + 1 = -1\). (Not distinct, \(x = y\))
• If \(x = 1\), then \(y = 2(1) + 1 = 3\). Since \(1\) and \(3\) are both in \(S\) and are distinct, this is a valid pair!

Let's verify: for \(x = 1\) and \(y = 3\):
\[
-|-2(1) + 3| = -|-2 + 3| = -|1| = -1
\]

• Case 2: \(y = 2x - 1\)
• If \(x = 1\), then \(y = 2(1) - 1 = 1\). (Not distinct)
• If \(x = -1\), then \(y = 2(-1) - 1 = -3\). Since \(-1\) and \(-3\) are both in \(S\) and are distinct, this is also a valid pair!

Let's verify: for \(x = -1\) and \(y = -3\):
\[
-|-2(-1) + (-3)| = -|2 - 3| = -|-1| = -1
\]

Determine the greatest value

The smallest possible positive integer value for the absolute value \(|-2x + y|\) is \(1\).
Therefore, the greatest value the expression \(-|-2x + y|\) can have is \(-1\).
Looking at the drop-down menu in the image, the selected value is \(3\), which is incorrect. The correct maximum value is \(-1\).
</reasoning>

<answer>
Choose different values for \(x\) and \(y\) from the integers \(-5, -3, -1, 0, 1, 3,\) and \(5\). What is the greatest value the expression below can have?

\(-|-2x + y|\) <blank>\(-1\)</blank>
</answer>

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"Absolute Value Optimization"
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Analyze the target expression

Using the Integer Operations knowledge point
The expression is:
\[
E = -|-2x + y|
\]
We want to find the greatest possible value of \(E\). Since the absolute value \(|-2x + y|\) is always non-negative (\(\ge 0\)), its negative \(-|-2x + y|\) must be non-positive (\(\le 0\)).
Thus, the maximum possible value of \(E\) is \(0\), which occurs if and only if:
\[
-2x + y = 0 \implies y = 2x
\]

Check for a zero value

We need to find distinct integers \(x\) and \(y\) from the set \(S = \{-5, -3, -1, 0, 1, 3, 5\}\) such that \(y = 2x\).
Let's test possible values for \(x \in S\):

• If \(x = -5\), then \(y = -10

otin S\).

• If \(x = -3\), then \(y = -6

otin S\).

• If \(x = -1\), then \(y = -2

otin S\).

• If \(x = 0\), then \(y = 0\). But \(x\) and \(y\) must be different values, so \(x

e y\). Thus, this pair is invalid.

• If \(x = 1\), then \(y = 2

otin S\).

• If \(x = 3\), then \(y = 6

otin S\).

• If \(x = 5\), then \(y = 10

otin S\).

Since no distinct pair satisfies \(y = 2x\), the expression \(E\) cannot equal \(0\).

Minimize the absolute value

To make \(E = -|-2x + y|\) as large as possible, we must make the absolute value \(|-2x + y|\) as small as possible. Since \(|-2x + y|\) cannot be \(0\), we look for distinct \(x, y \in S\) that make \(|-2x + y|\) equal to \(1\), \(2\), or the smallest possible positive integer.

Let's test if we can get \(|-2x + y| = 1\):
This means \(-2x + y = 1\) or \(-2x + y = -1\).

• Case 1: \(y = 2x + 1\)
• If \(x = -1\), then \(y = 2(-1) + 1 = -1\). (Not distinct, \(x = y\))
• If \(x = 1\), then \(y = 2(1) + 1 = 3\). Since \(1\) and \(3\) are both in \(S\) and are distinct, this is a valid pair!

Let's verify: for \(x = 1\) and \(y = 3\):
\[
-|-2(1) + 3| = -|-2 + 3| = -|1| = -1
\]

• Case 2: \(y = 2x - 1\)
• If \(x = 1\), then \(y = 2(1) - 1 = 1\). (Not distinct)
• If \(x = -1\), then \(y = 2(-1) - 1 = -3\). Since \(-1\) and \(-3\) are both in \(S\) and are distinct, this is also a valid pair!

Let's verify: for \(x = -1\) and \(y = -3\):
\[
-|-2(-1) + (-3)| = -|2 - 3| = -|-1| = -1
\]

Determine the greatest value

The smallest possible positive integer value for the absolute value \(|-2x + y|\) is \(1\).
Therefore, the greatest value the expression \(-|-2x + y|\) can have is \(-1\).
Looking at the drop-down menu in the image, the selected value is \(3\), which is incorrect. The correct maximum value is \(-1\).
</reasoning>

<answer>
Choose different values for \(x\) and \(y\) from the integers \(-5, -3, -1, 0, 1, 3,\) and \(5\). What is the greatest value the expression below can have?

\(-|-2x + y|\) <blank>\(-1\)</blank>
</answer>

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