is |(x + 3)||(e^x)| + |-6||e^2| greater than this : |(x + 3)(e^x) + -6e^2|

To determine whether the expression:

\[
|(x + 3)| \cdot |e^x| + |-6| \cdot |e^2|
\]

is greater than:

\[
|(x + 3)(e^x) + (-6e^2)|
\]

we need to analyze the expressions more closely.

### Expression 1: \( |(x + 3)| \cdot |e^x| + |-6| \cdot |e^2| \)

This expression involves the sum of two products:

1. \( |(x + 3)| \cdot |e^x| \)
2. \( |-6| \cdot |e^2| \)

Since the exponential function \( e^x \) and \( e^2 \) are always positive, the absolute values simplify to:

\[
|x + 3| \cdot e^x + 6e^2
\]

### Expression 2: \( |(x + 3)(e^x) + (-6e^2)| \)

This is the absolute value of a single expression:

\[
|(x + 3) \cdot e^x - 6e^2|
\]

### Key Consideration: When Is Expression 1 Greater Than Expression 2?

We want to know if:

\[
|x + 3| \cdot e^x + 6e^2 > |(x + 3) \cdot e^x - 6e^2|
\]

#### Analyzing the Difference

Let's define:

\[
A = |x + 3| \cdot e^x + 6e^2
\]

\[
B = |(x + 3) \cdot e^x - 6e^2|
\]

We need to determine when \( A > B \).

The expression \( A \) involves adding two non-negative terms, whereas \( B \) is the absolute value of a potentially positive or negative quantity. Notice that if \( (x + 3) \cdot e^x \) is large enough compared to \( 6e^2 \), then \( B \) could be smaller than \( A \).

### Conclusion

In general, **yes**, the expression \( |(x + 3)| \cdot e^x + 6e^2 \) is greater than \( |(x + 3) \cdot e^x - 6e^2| \) because \( A \) involves a straightforward sum of positive quantities, while \( B \) takes the absolute value, potentially reducing its size.

Would you like more details or have additional questions?

Here are 5 related questions you might find interesting:
1. What is the importance of absolute values in expressions like these?
2. How does the exponential function \( e^x \) behave as \( x \) increases or decreases?
3. Can you always distribute absolute values over addition and multiplication?
4. How do you determine when one expression is greater than another involving variables?
5. What are some common properties of exponential functions like \( e^x \)?

**Tip:** When comparing expressions with absolute values, consider breaking them down into cases based on the sign of the expressions inside the absolute value!

Explore the comparison of absolute values and exponential functions inequalities. Understand when the expression involving absolute values of products is greater than another expression involving products and sums. Detailed mathematical analysis provided.

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