Question

a triangle has two sides of length 4.7 and 3.7. what compound inequality describes the possible lengths for the third side, x? write a compound inequality like 1 < x < 3.

Explanation:

Step1: Apply triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side.
Let the two given side - lengths be \(a = 4.7\) and \(b = 3.7\).
The difference between the two sides is \(|a - b|=|4.7−3.7| = 1\).
The sum of the two sides is \(a + b=4.7 + 3.7=8.4\).

Step2: Write the compound inequality

Based on the triangle - inequality theorem, the possible lengths of the third side \(x\) satisfy the compound inequality \(|a - b|\lt x\lt a + b\).
So, \(1\lt x\lt8.4\).

Answer:

\(1\lt x\lt8.4\)