Geometry

Question Analysis

The main focus is to use the Triangle Inequality Theorem to find the range of values for given the side lengths of a triangle. We need to establish inequalities based on the theorem and solve them to determine the valid values of .

 

Key Concept Explanation

The Triangle Inequality Theorem states that for a triangle with side lengths , , and , the following three inequalities must hold: a + b>c, a + c>b, and b + c>a. In this case, the side lengths are , , and . We will use these inequalities to find the range of .

 

Step - by - Step Solution

1. First inequality: 8+11>3x - 5

Simplify the left - hand side: 19>3x - 5.

Add to both sides: 19 + 5>3x, so 24>3x.

Divide both sides by : x < 8.

2. Second inequality: 8+(3x - 5)>11

Simplify the left - hand side: 8+3x - 5>11, which becomes 3x+3>11.

Subtract from both sides: 3x>11 - 3, so 3x>8.

Divide both sides by : x>

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