Question

multiple choice
which two of the following compound inequalities simplify?
a x < 3 or x > 5
b x < 3 and x > 5
c x < 3 or x < 5
d x < 3 and x < 5

Explanation:

Step1: Analyze Option A

The compound inequality is \( x < 3 \) or \( x > 5 \). There is no overlap between the two intervals, and it cannot be simplified further as "or" combines non - overlapping regions. But we will check other options too.

Step2: Analyze Option B

The compound inequality is \( x < 3 \) and \( x > 5 \). A number cannot be both less than 3 and greater than 5 at the same time. So this inequality has no solution, and it doesn't simplify in a meaningful way to a non - empty interval.

Step3: Analyze Option C

The compound inequality is \( x < 3 \) or \( x < 5 \). If a number is less than 3, it is also less than 5. So the "or" of \( x < 3 \) and \( x < 5 \) simplifies to \( x < 5 \) because all numbers less than 3 are already included in the set of numbers less than 5.

Step4: Analyze Option D

The compound inequality is \( x < 3 \) and \( x < 5 \). If a number is less than 3, it is also less than 5. So the "and" of \( x < 3 \) and \( x < 5 \) simplifies to \( x < 3 \) because the intersection of the set of numbers less than 3 and the set of numbers less than 5 is the set of numbers less than 3.

Answer:

C. \( x < 3 \) or \( x < 5 \), D. \( x < 3 \) and \( x < 5 \)