Subtraction of inequalities Suppose I have two inequalities:
$$k_1 \leq a$$ 
$$k_2 \leq b$$

where $k_1,k_2,a,b$ are all positive numbers

I know that that summation of them can be written as:
$$k_1+k_2 \leq a+b$$

But I want to find $$k_1-k_2 \leq ?$$ or $$? \leq k_1-k_2$$

Is there any way to solve this problem?
First thing you can do is invoke the triangle inequality, i.e. $|x + y| \leq |x| + |y|$. Note that this equality works for negative $x$, $y$ as well. This gives you $$
\begin{eqnarray}
|k_1 - k_2| &amp;\leq&amp; |k_1| + |k_2| &amp;\leq |a| + |b| \text{ and since }k_1-k_2 \leq |k_1-k_2| \\
k_1 - k_2 &amp;\leq&amp; |k_1| + |k_2|  &amp;\leq |a| + |b|
\end{eqnarray}
$$

For arbitrary $k_1 \leq a$, $k_2 \leq b$, you can't do better than that, but since you stated that $k_1,k_2 \geq 0$, you can. $k_1,k_2$ being positive means that $$
\begin{eqnarray}
-b &amp;\leq&amp; -k_2 &amp;\leq&amp; k_1 - k_2 &amp;\leq&amp; k_1 &amp;\leq&amp; a\\
-a &amp;\leq&amp; -k_1 &amp;\leq&amp; k_2 - k_1 &amp;\leq&amp; k_2 &amp;\leq&amp; b
\end{eqnarray}
$$
because subtracting a positive number can only make a number smaller, never bigger.
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 &lt; d from a &lt; b yields a - d &lt; 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)  &lt;-----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