Question

in order for a gear to work in a piece of machinery, the radius of the gear, ( r ), must be greater than 4 cm, but not exceed 4.1 cm. which compound inequality represents the situation?

( r < 4 ) or ( r geq 4.1 )

( r > 4 ) or ( r leq 4.1 )

( r > 4 ) and ( r leq 4.1 )

( r < 4 ) and ( r geq 4.1 )

Explanation:

Step1: Analyze the first condition

The radius \( r \) must be greater than 4 cm. In inequality form, this is \( r > 4 \).

Step2: Analyze the second condition

The radius \( r \) must not exceed 4.1 cm, which means \( r \) is less than or equal to 4.1 cm. In inequality form, this is \( r \leq 4.1 \).

Step3: Combine the conditions

Since both conditions must be true at the same time (the radius has to satisfy both being greater than 4 and not exceeding 4.1), we use the "and" connector. So the compound inequality is \( r > 4 \) and \( r \leq 4.1 \).

Answer:

\( r > 4 \) and \( r \leq 4.1 \) (the third option among the given choices)