Question

writing a compound inequality
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 this situation?
\\( r > 4 \\) and \\( r \leq 4.1 \\)
\\( r > 4 \\) or \\( r \leq 4.1 \\)
\\( r < 4 \\) and \\( r \geq 4.1 \\)
\\( r < 4 \\) or \\( r \geq 4.1 \\)

Explanation:

Step1: Analyze "greater than 4"

The radius \( r \) must be greater than 4 cm, so we have the inequality \( r > 4 \).

Step2: Analyze "not exceed 4.1"

"Not exceed 4.1" means \( r \) is less than or equal to 4.1 cm, so the inequality is \( r \leq 4.1 \).

Step3: Determine the compound inequality type

Since \( r \) must satisfy both conditions (greater than 4 AND not exceed 4.1) simultaneously, we use "and" to combine the two inequalities. So the compound inequality is \( r > 4 \) and \( r \leq 4.1 \).

Answer:

A. \( r > 4 \) and \( r \leq 4.1 \)