Use the image to answer the question.

Question

A number line with arrows at both the ends ranges from negative 10 to 10 in unit increments. A closed point is marked on negative 1. A leftward arrow extends along the number line from the closed point toward negative 10 and beyond.

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

(1 point)
Responses

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 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 .

an infinite number
an infinite number

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

Answer 1

The inequality \( x \leq -1 \) indicates that any value less than or equal to -1 is a solution. To find how many solutions exist between -4 and -5, we can consider the decimal values within that range.

The interval of interest is from -4 to -5, which includes all decimal values from \( -4.001 \) to \( -4.999 \).

1. Counting Solutions:
   • From -4.001 to -4.999, we observe that between any two decimal numbers, there are infinitely many more decimal numbers. For example, between -4.001 and -4.002, you can have -4.0015, -4.0011, etc.

Therefore, since there are infinitely many decimal numbers within that interval, the correct response is:

an infinite number