Question

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a triangle has side lengths of 200 units and 300 units. write a compound inequality for the range of the possible lengths for the third side, x.

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side, and the absolute difference of the lengths of any two sides must be less than the length of the remaining side. For a triangle with sides \(a\), \(b\), and \(x\) (the third side), we have \(|a - b| < x < a + b\).

Step2: Identify the given sides

Here, the two given sides are \(a = 300\) units and \(b = 200\) units. First, calculate the difference: \(|300 - 200| = 100\). Then, calculate the sum: \(300 + 200 = 500\).

Step3: Form the compound inequality

Using the triangle inequality theorem, the length of the third side \(x\) must satisfy \(100 < x < 500\).

Answer:

\(100 < x < 500\)