Answer: Choice D
19/12
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Explanation:
Use the triangle inequality theorem to set up these three inequalities
- AB+BC > AC
- AB+AC > BC
- AC+BC > AB
If you were to apply substitution and solve for x, then you'd get these solutions in the order presented shown above
We'll ignore the third inequality. It's not very useful because we were already told that x > 0.
The first two inequalities combine to get 3/4 < x < 11/2
This is the range of possible x values so that triangle ABC is possible.
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If we want angle CAB (or angle A for short) to be the largest of the triangle, then side BC must be the longest of the three sides.
The rule is: the largest angle is always opposite the longest side. This applies to triangles only.
Since BC is the longest side, this means BC > AB and BC > AC
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Let's focus on the first inequality
BC > AB
x+7 > x+4
7 > 4
That's true regardless of what we pick for x. So BC is always longer than AB.
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Now onto the second inequality
BC > AC
x+7 > 4x
7 > 4x-x
7 > 3x
3x < 7
x < 7/3
Now recall that we found earlier x > 3/4 or 3/4 < x
Combine 3/4 < x and x < 7/3 to get the compound inequality 3/4 < x < 7/3
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Notice the middle part of that compound inequality is x and BC = x+7
If we were to add 7 to all three parts of the compound inequality, then we'll determine exactly the possible range of lengths for segment BC.
3/4 < x < 7/3
3/4+7 < x+7 < 7/3+7
3/4 + 28/4 < x+7 < 7/3 + 21/3
31/4 < x+7 < 28/3
31/4 < BC < 28/3
The length of segment BC is between 31/4 units and 28/3 units, excluding both endpoints. Those are the values of m and n in that order.
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The last step is to compute n-m
n-m = (28/3) - (31/4)
n-m = (112/12) - (93/12)
n-m = (112-93)/12
n-m = 19/12 which points us to choice D.
Side note: 7/3 - 3/4 = 28/12 - 9/12 = (28-9)/12 = 19/12
