To determine if three lengths can form a triangle, we can use the triangle inequality theorem. The theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must be satisfied:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's denote the sides of the triangle as \( a = 2 \), \( b = 4 \), and \( c = 7 \). We will check each inequality:
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\( a + b > c \): \[ 2 + 4 > 7 \implies 6 > 7 \quad \text{(False)} \]
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\( a + c > b \): \[ 2 + 7 > 4 \implies 9 > 4 \quad \text{(True)} \]
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\( b + c > a \): \[ 4 + 7 > 2 \implies 11 > 2 \quad \text{(True)} \]
Since the first inequality is false, the lengths 2, 4, and 7 cannot form a triangle.