Question
A number line is shown extending from -10 to 10, an arrow starts from -1 to beyond -10, the arrow goes through and between -4 and -5, the number line is showing solutions to the inequality x <= -1. if you could use decimals, how many solutions are there between -4 and -5
Answers
GPT-4o mini
The inequality \( x \leq -1 \) represents all numbers that are less than or equal to -1. This includes numbers from \(-10\) (or lower) up to \(-1\).
The segment between \(-4\) and \(-5\) contains all the decimal numbers from \(-5\) to \(-4\). Specifically, these numbers can be written as:
\[
x \in (-5, -4)
\]
We are interested in the solutions within this segment that also satisfy \( x \leq -1 \). Since \(-4\) and \(-5\) are both less than \(-1\), all the decimal numbers in the range from -5 (not including -5) to -4 (not including -4) are valid solutions.
The number of solutions in this interval is infinite. You can think of every possible decimal value between \(-5\) and \(-4\) (such as \(-4.1\), \(-4.01\), \(-4.001\), etc.), thereby giving an infinite number of solutions as well.
In conclusion, there are infinitely many solutions to the inequality \( x \leq -1 \) in the interval between \(-4\) and \(-5\).
The segment between \(-4\) and \(-5\) contains all the decimal numbers from \(-5\) to \(-4\). Specifically, these numbers can be written as:
\[
x \in (-5, -4)
\]
We are interested in the solutions within this segment that also satisfy \( x \leq -1 \). Since \(-4\) and \(-5\) are both less than \(-1\), all the decimal numbers in the range from -5 (not including -5) to -4 (not including -4) are valid solutions.
The number of solutions in this interval is infinite. You can think of every possible decimal value between \(-5\) and \(-4\) (such as \(-4.1\), \(-4.01\), \(-4.001\), etc.), thereby giving an infinite number of solutions as well.
In conclusion, there are infinitely many solutions to the inequality \( x \leq -1 \) in the interval between \(-4\) and \(-5\).