Finding the inequality with k(1) - K(2) as the subject
The simplest solution is usually the best solution----Albert Einstein
Given:
k(1) ≤ a
k(2) ≤ b
Required: To determine if k(1) - K(2) can be made the subject of an inequality
derived from the above inequalities
Possible solution:
One will operate on the given system of inequalities.
Generally, subtracting c < d from a < b yields a - d < b - c; but below, one
will apply the permissible operations on a system of inequalities. To add a
system of two inequalities, both inequalities must have the same sense (or
direction). To subtract an inequality, multiply this "subtrahend" inequality by
-1 while reversing the sense of the inequality, and then add to the "minuend
inequality", making sure the two inequalities have the same sense.
Now, one will subtract K(2) ≤ b from k(1) ≤ a).
Step 1: Multiply the "subtrahend" inequality, K(2) ≤ b by -1 and reverse the
sense to obtain - K(2) ≥ - b......(1)
Step 2: Rewrite (1) so that it has the same sense as the "minuend" inequality,
k(1) ≤ a. Then one obtains -b ≤ -k(2)
Step 3 Add the left sides and add the right sides below
k(1) ≤ a -b ≤ -K(2) ---------------
k(1) - b ≤ a - k(2) <-----the "difference" inequality,
From the above result, k(1) - k(2) cannot be made the subject of the above
inequality.
Conclusion:
Therefore, explicitly, k(1) - k(2) does not exist.
However, implicitly, k(1) - k(2) ≤ a + b - 2k(2)
Repeating the above procedure, using a, b, c, d
Given
a ≤ b...........(1).
c ≤ d............(2);
determine if a - c can be made the subject of the inequality derived from the above system of inequalities.
Solution
From (2) -c ≥ -d (multiplying by -1 in order to subtract)
-d ≤ -c....(3) (rewriting in the same sense as (1))
Adding the left sides of (1) and (3); and adding the right sides of (1) and (3),
one obtains
a - d ≤ b - c...........(4)
Conclusion
Explicitly, in (4), a - c cannot be made the subject of (4),
However, implicitly, a - c ≤ b + d - 2c