2n + 19 = 2( ) + 1Let n be an integer. Use Definition 1.6 to explain why 
2n + 19
 is an odd integer.

To solve this and explain why \( 2n + 19 \) is an odd integer, let's use **Definition 1.6**, which likely defines **odd integers**.

An **odd integer** is typically defined as any integer of the form:

\[
\text{Odd integer} = 2k + 1
\]
where \( k \) is an integer.

### Step 1: Understand the given equation
The equation provided is:
\[
2n + 19 = 2( \ ) + 1
\]
We are tasked with proving that \( 2n + 19 \) is odd.

### Step 2: Break it down
We know that for an integer to be odd, it must take the form \( 2k + 1 \). Let's express \( 2n + 19 \) in a form that matches this definition.

We can rewrite \( 19 \) as \( 2 \times 9 + 1 \), i.e.:
\[
19 = 2 \times 9 + 1
\]

Now substitute this into the original equation:
\[
2n + 19 = 2n + (2 \times 9 + 1) = 2(n + 9) + 1
\]

### Step 3: Explanation
From this, we can see that:
\[
2n + 19 = 2(n + 9) + 1
\]
This matches the form of an odd integer \( 2k + 1 \), where \( k = n + 9 \), which is also an integer because \( n \) is an integer by assumption.

Thus, **\( 2n + 19 \) is an odd integer** because it can be written in the form \( 2k + 1 \), where \( k = n + 9 \), and \( k \) is an integer.

### Conclusion
By **Definition 1.6**, \( 2n + 19 \) is an odd integer, since it is of the form \( 2k + 1 \) with \( k = n + 9 \).

---

Would you like further details or have any questions?

Here are 5 related questions to expand this topic:
1. What are the characteristics of even and odd numbers?
2. How do we prove that the sum of two odd integers is even?
3. Can the expression \( 2n + 19 \) ever be even?
4. How does the concept of divisibility apply to even and odd numbers?
5. What other forms can be used to represent odd integers?

**Tip**: When proving a number is odd, always try to express it in the form \( 2k + 1 \), where \( k \) is an integer.

2n + 19 = 2( ) + 1. Let n be an integer. Use Definition 1.6 to explain why 2n + 19 is an odd integer.

Learn how to prove that 2n + 19 is an odd integer using Definition 1.6. This problem explores the algebraic structure of odd numbers and uses step-by-step reasoning to express the given expression in the form 2k + 1. Suitable for students in Grades 8-10.

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