The image shows solutions to the inequality x≤−1 . How many solutions are there between −4 and −5 ?

Question

The image shows solutions to the inequality  x≤−1 . How many solutions are there between  −4  and  −5 ?

(1 point)
Responses

There are 999 solutions: all the decimals from −4.001 to −4.999.
There are 999 solutions: all the decimals from negative 4.001 to negative 4.999 .

There are 99 solutions: all the decimals from −4.01 to −4.99.
There are 99 solutions: all the decimals from negative 4.01 to negative 4.99 .

There are 9,999 solutions: all the decimals from −4.0001 to −4.9999. 
There are 9,999 solutions: all the decimals from negative 4.0001 to negative 4.9999 .

an infinite number

GPT-4o mini · Answer

The inequality $ x \leq -1 $ indicates that $ x $ can take any value less than or equal to $ -1 $. Given the range between $-4$ and $-5$, we can analyze this interval: - The interval from $-4$ to $-5$ is actually from the larger number $-4$ (not including $-5$ since the interval does not specify it) down to just above $-5$. Thus, the numbers in this range can be expressed as all values in the form of decimals (e.g., $-4.0001, -4.0002, ...$ down to values approaching $-5$), which is a continuous range of numbers. Since there are infinitely many decimal numbers in any interval on the real number line, between any two real numbers (even as close as $-4$ and $-5$), there are infinitely many solutions. Therefore, the correct answer is: **an infinite number**.