To determine which pairs of lengths can be the other two sides of a triangle with the third side being \( x \) (as indicated by the range on the number line), we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given that \( x \) is the length of the third side, our possible pairs of side lengths can be denoted as \( a \) and \( b \). The inequalities we need to satisfy for side lengths \( a \) and \( b \) with the third side \( x \) are:
- \( a + b > x \)
- \( a + x > b \)
- \( b + x > a \)
For each pair of lengths you've provided, we would check whether they satisfy these inequalities within the bounds of \( x \) as indicated on the number line.
However, since the numerical range for \( x \) isn't given in your question, let’s analyze the options using hypothetical limits:
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42 inches and 50 inches: \( a = 42 \), \( b = 50 \)
- Check \( a + b > x \): \( 42 + 50 = 92 \) should be greater than \( x \)
- Check \( a + x > b \): \( 42 + x > 50 \) (x must be > 8)
- Check \( b + x > a \): \( 50 + x > 42 \) (x must be > -8)
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28 inches and 64 inches: \( a = 28 \), \( b = 64 \)
- \( 28 + 64 = 92 > x \)
- \( 28 + x > 64 \) (x must be > 36)
- \( 64 + x > 28 \) (x must be > -36)
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36 inches and 92 inches: \( a = 36 \), \( b = 92 \)
- \( 36 + 92 = 128 > x \)
- \( 36 + x > 92 \) (x must be > 56)
- \( 92 + x > 36 \) (x must be > -56)
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48 inches and 76 inches: \( a = 48 \), \( b = 76 \)
- \( 48 + 76 = 124 > x \)
- \( 48 + x > 76 \) (x must be > 28)
- \( 76 + x > 48 \) (x must be > -28)
Determine the values of \( x \) based on the constraints and compare those with the given ranges. As I don't have the specific range from the number line, you will need to analyze if any of these pairs fall within your defined \( x \) range, using the inequalities provided.
If you can provide the specific range from the number line, I can help you identify the correct option directly!
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