Question

which set of numbers is included as part of the solution set of the compound inequality ( x < 6 ) or ( x > 10 )?
( circ {-7, -1.7, 6.1, 10} )
( circ {-3, 4.5, 13.6, 19} )
( circ {0, 6, 9.8, 14} )
( circ {8.5, 9.1} )

Explanation:

Step1: Analyze the compound inequality

The compound inequality is \( x < 6 \) or \( x > 10 \). We need to check each number in the sets to see if it satisfies either of these inequalities.

Step2: Check the first set \(\{-7, -1.7, 6.1, 10\}\)

• \(-7\): \(-7 < 6\), satisfies \( x < 6 \).
• \(-1.7\): \(-1.7 < 6\), satisfies \( x < 6 \).
• \(6.1\): \(6.1\) is not less than \(6\) and not greater than \(10\) (\(6.1 < 10\)), so does not satisfy either.
• \(10\): \(10\) is not less than \(6\) and not greater than \(10\) (\(10 = 10\)), so does not satisfy either. Since there are numbers that don't satisfy, this set is not part of the solution.

Step3: Check the second set \(\{-3, 4.5, 13.6, 19\}\)

• \(-3\): \(-3 < 6\), satisfies \( x < 6 \).
• \(4.5\): \(4.5 < 6\), satisfies \( x < 6 \).
• \(13.6\): \(13.6 > 10\), satisfies \( x > 10 \).
• \(19\): \(19 > 10\), satisfies \( x > 10 \). All numbers satisfy either \( x < 6 \) or \( x > 10 \).

Step4: Check the third set \(\{0, 6, 9.8, 14\}\)

• \(0\): \(0 < 6\), satisfies \( x < 6 \).
• \(6\): \(6\) is not less than \(6\) and not greater than \(10\) (\(6 = 6\)), so does not satisfy either.
• \(9.8\): \(9.8\) is not less than \(6\) and not greater than \(10\) (\(9.8 < 10\)), so does not satisfy either.
• \(14\): \(14 > 10\), satisfies \( x > 10 \). Since there are numbers that don't satisfy, this set is not part of the solution.

Step5: Check the fourth set \(\{8.5, 9.1\}\)

• \(8.5\): \(8.5\) is not less than \(6\) and not greater than \(10\) (\(8.5 < 10\)), so does not satisfy either.
• \(9.1\): \(9.1\) is not less than \(6\) and not greater than \(10\) (\(9.1 < 10\)), so does not satisfy either. This set is not part of the solution.

Answer:

\(\{-3, 4.5, 13.6, 19\}\) (the second set)