Which statements describe the solutions to the inequality [tex]\(x \ \textless \ -20\)[/tex]? Check all that apply.

A. There are infinite solutions.
B. Each solution is negative.
C. Each solution is positive.
D. The solutions are both positive and negative.
E. The solutions contain only integer values.
F. The solutions contain rational number values.



Answer :

Let's examine the inequality [tex]\( x < -20 \)[/tex] and determine which statements correctly describe its solutions.

1. There are infinite solutions:
- The inequality [tex]\( x < -20 \)[/tex] includes all numbers that are less than -20. This set of numbers continues indefinitely to the left on the number line. Therefore, there are infinitely many values that satisfy this inequality.
- Conclusion: This statement is true.

2. Each solution is negative:
- Any number that is less than -20 is negative because -20 itself and any smaller number than -20 (like -21, -22, -100, etc.) are all below zero on the number line.
- Conclusion: This statement is true.

3. Each solution is positive:
- Numbers that are less than -20 are all negative because -20 is a negative number, and any number smaller than a negative number remains negative. Hence, none of these solutions can be positive.
- Conclusion: This statement is false.

4. The solutions are both positive and negative:
- Since all numbers less than -20 are negative, there are no positive numbers among the solutions. Therefore, the set of solutions cannot include both positive and negative numbers.
- Conclusion: This statement is false.

5. The solutions contain only integer values:
- The inequality [tex]\( x < -20 \)[/tex] includes all numbers less than -20, not just integers. This set includes fractions, decimals, and any other rational numbers. For instance, -20.5, -21.1, and -30.33 are also valid solutions.
- Conclusion: This statement is false.

6. The solutions contain rational number values:
- Rational numbers include all integers, fractions, and decimals that can be expressed as a ratio of two integers. Since [tex]\( x < -20 \)[/tex] includes all these types of numbers as long as they are less than -20, the solutions definitely contain rational number values.
- Conclusion: This statement is true.

Summary of Correct Statements:
- There are infinite solutions.
- Each solution is negative.
- The solutions contain rational number values.