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.
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.
