- #1
silvermane
Gold Member
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1. The problem statement:
Suppose that a, b, x, and y are real numbers satisfying
a < x < b and a < y < b.
Show that |x - y| < b - a.
We may start with the fact that:
i.) a < x < b
ii.) a < y < b
Subtracting a from all sides, we have that
i.) 0 = a - a < x - a < b - a
ii.) 0 = a - a < y - a < b - a
and
0 < x - a < b - a
0 < y - a < b - a
but I'm stuck here in justifying my steps. I thought about subtracting the inequalities from each other but that would just yield
0 < x - y < 0
which doesn't help my case :(
Any help and/or tips would be lovely!
Suppose that a, b, x, and y are real numbers satisfying
a < x < b and a < y < b.
Show that |x - y| < b - a.
The Attempt at a Solution
We may start with the fact that:
i.) a < x < b
ii.) a < y < b
Subtracting a from all sides, we have that
i.) 0 = a - a < x - a < b - a
ii.) 0 = a - a < y - a < b - a
and
0 < x - a < b - a
0 < y - a < b - a
but I'm stuck here in justifying my steps. I thought about subtracting the inequalities from each other but that would just yield
0 < x - y < 0
which doesn't help my case :(
Any help and/or tips would be lovely!