Answer
The valid lengths for possible values of b are 0.5 in. and 2 in. To assess the acceptable values for the base b of an isosceles triangle with 2a + b = 15.7 inches and ensuring that the sum of any two sides exceeds the length of the third side, we apply the triangle inequality theorem. Following the conditions: 2a + b = 15.7, we have a + a > b (hence, 2a > b), a + b > a (which gives b > 0), and b + a > a (which again leads to b > 0). We should evaluate the options: –2 in.: Invalid as –2 is negative; b must be positive. 0 in.: Invalid since a base cannot be of zero length. 0.5 in.: Valid because if 2a = 15.7 – 0.5 = 15.2, then 2a > 0.5 and a > 0 hold true. 2 in.: Valid as 2a = 15.7 – 2 = 13.7 and 2a > 2, with a > 0 satisfied. 7.9 in.: Valid if 2a = 15.7 – 7.9 = 7.8, leading to 2a > 7.9 and a > 0 being valid. Thus, according to the triangle inequality theorem, the allowable options for b are 0.5 in. and 2 in.
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