Carlos is using a number line to add another integer to –4. He begins by showing –4 on the number line, as shown below. Carlos adds the other integer on the number line and says that the sum is negative.

Question

A number line going from negative 5 to positive 5. The area between negative 4 to 0 is highlighted.

Which is true of the number Carlos added to –4?
The number must be a positive integer.
The number could be 3 or any integer less than 3.
The number could be 4 or any integer greater than 4.

Answer 1

To determine which statement is true regarding the number Carlos added to -4 that still results in a negative sum, we need to analyze the implications of the sum being negative.

The sum \( -4 + x \) is negative if:

\[ -4 + x < 0 \]

Solving this inequality, we add 4 to both sides:

\[ x < 4 \]

This means that the number Carlos added (denoted as \( x \)) must be less than 4.

Now, let's evaluate the given options based on this finding:

1. The number must be a positive integer.

   • This is not necessarily true. \( x \) could be anything less than 4, which includes negative integers and numbers like 0, 1, 2, or 3, so this option is false.

2. The number could be 3 or any integer less than 3.

   • This is true. The number can be 3, 2, 1, 0, -1, -2, etc. All these numbers are less than 4 and would keep the sum negative.

3. The number could be 4 or any integer greater than 4.

   • This is false, as adding 4 or any number greater than 4 would make the sum zero or positive (e.g., -4 + 4 = 0 or -4 + 5 = 1).

Therefore, the correct answer is:

The number could be 3 or any integer less than 3.