Question

Given \( A = |x+3 + |x-2| - |2x-8| \). The maximum value of \( |A| \) is : 

• A. 111
• B. 9
• C. 6
• D. 3
• E. ∞

Answer 1

The Correct Option is B

To find the maximum value of \( |A| \), where \( A = |x+3 + |x-2| - |2x-8| \), we need to carefully analyze the behavior of the expression as a function involving absolute values.

Let's break down the function by considering critical points derived from the absolute value expressions:

1. Critical points occur where the inside of the absolute values is zero:
   • \( x+3 = 0 \Rightarrow x = -3 \)
   • \( x-2 = 0 \Rightarrow x = 2 \)
   • \( 2x-8 = 0 \Rightarrow x = 4 \)

Using these points, we examine intervals defined by the points \( x = -3, 2, 4 \). Study each interval separately:

1. For \( x < -3 \):
   • \( |x+3| = -(x+3) \)
   • \( |x-2| = -(x-2) \)
   • \( |2x-8| = -(2x-8) \)
2. For \( -3 \leq x < 2 \):
   • \( |x+3| = x+3 \)
   • \( |x-2| = -(x-2) \)
   • \( |2x-8| = -(2x-8) \)
3. For \( 2 \leq x < 4 \):
   • \( |x+3| = x+3 \)
   • \( |x-2| = x-2 \)
   • \( |2x-8| = -(2x-8) \)
4. For \( x \geq 4 \):
   • \( |x+3| = x+3 \)
   • \( |x-2| = x-2 \)
   • \( |2x-8| = 2x-8 \)

Calculating the value of \( |A| \) in these cases reveals that the maximum value occurs at intervals where \( A = 9 \). Therefore, the maximum value of \( |A| \) is \(9\).

Given these calculations, the correct answer is 9.

Answer 2

We want to find the maximum value of \(|A|\). For this, we analyze each absolute value term separately. 

Step 1: Break down each absolute value expression. 

• \(|x+3|\): \[ |x+3| = \begin{cases} x + 3, & x \geq -3 \\ -x - 3, & x < -3 \end{cases} \]
• \(|x-2|\): \[ |x-2| = \begin{cases} x - 2, & x \geq 2 \\ -x + 2, & x < 2 \end{cases} \]
• \(-|2x-8|\): \[ -|2x-8| = \begin{cases} -(2x - 8), & x \leq 4 \\ (2x - 8), & x > 4 \end{cases} \]

Step 2: Analyze different ranges of \(x\).

Case 1: \(x \geq 4\)

\[ A = (x+3) + (x-2) - (2x - 8) = 7 \]

Case 2: \(2 \leq x < 4\)

\[ A = (x+3) + (x-2) - (-2x + 8) = 4x - 7 \]

Case 3: \(-3 \leq x < 2\)

\[ A = (x+3) - (x-2) - (-2x + 8) = 2x - 3 \]

Case 4: \(x < -3\)

\[ A = (-x - 3) + (-x + 2) - (-2x + 8) = -9 \]

Step 3: Evaluate maximum absolute value.

• Case 1: \(A = 7 \implies |A| = 7\)
• Case 2: \(A = 4x - 7\). At \(x = 2 \implies A = 1\), at \(x = 4 \implies A = 9\). So maximum is 9.
• Case 3: \(A = 2x - 3\). At \(x = -3 \implies A = -9\), at \(x = 2 \implies A = 1\). So maximum absolute value is 9.
• Case 4: \(A = -9 \implies |A| = 9\)

Step 4: Conclusion

\[ \max |A| = 9 \]

Final Answer:

Option B is the correct answer.