If `5 < x < 10` and `y = x + 5`, what is the greatest possible integer value of `x + y`?

Options

A. 18
B. 20
C. 23
D. 24

Answer

To solve the problem step by step, we start with the given inequalities and the expression for \( y \).

### Step 1: Understand the given inequalities
We are given:
\[ 5 < x < 10 \]

### Step 2: Express \( y \) in terms of \( x \)
We know that:
\[ y = x + 5 \]

### Step 3: Find the range of \( y \)
Since \( y = x + 5 \), we can add 5 to the entire inequality of \( x \):
\[ 5 < x < 10 \implies 5 + 5 < x + 5 < 10 + 5 \]
This simplifies to:
\[ 10 < y < 15 \]

### Step 4: Find the expression for \( x + y \)
Now, we want to find \( x + y \):
\[ x + y = x + (x + 5) = 2x + 5 \]

### Step 5: Determine the range of \( x + y \)
We need to find the range of \( 2x + 5 \). First, we will find the minimum and maximum values of \( 2x + 5 \) using the bounds of \( x \).

1. **Minimum value of \( x + y \)**:
- When \( x \) is at its minimum (just above 5), we can take \( x \) to be slightly more than 5:
\[ 2(5) + 5 = 10 + 5 = 15 \]

2. **Maximum value of \( x + y \)**:
- When \( x \) is at its maximum (just below 10), we can take \( x \) to be slightly less than 10:
\[ 2(10) + 5 = 20 + 5 = 25 \]

Thus, we have:
\[ 15 < x + y < 25 \]

### Step 6: Identify the greatest possible integer value
The greatest possible integer value of \( x + y \) that is less than 25 is:
\[ 24 \]

### Final Answer
The greatest possible integer value of \( x + y \) is \( \boxed{24} \).

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