We can’t reverse either of these implications and still end up with a valid statement. If I’m thinking of a
rational number, I’m not necessarily thinking of a natural number. (Suppose it’s 1/2.) If I’m thinking of a
real number, I’m not necessarily thinking of a rational number. (Suppose it’s the positive square root of 2.)
Logical implication is not an equivalence relation, because it’s not symmetric.
Inequality Morphing
Some of the familiar equation-morphing rules also work for inequalities, but others must be
modified, and a few don’t work at all. Here are the things we can do with two-part equations,
summarized for reference.
- Reverse the order.
- Add the same quantity to both sides.
- Subtract the same quantity from both sides.
- Add one equation to another.
- Multiply both sides by the same quantity.
- Divide both sides by the same nonzero quantity.
Whenever we do one or more of these things to an equation, we get another valid equation.
Now let’s see how well these rules work for inequalities. (We won’t get into formal proofs of
these facts.) You can try out some examples if you want to improve your understanding of
how they work.
Manipulatingñ statements
If two quantities a and b are different, we can express that fact in either order. In general, it is
always true that if a≠b, then b≠a.
We can add or subtract the same quantity from each side of a “not equal” statement. If
two quantities are different to start out with, then they’ll still be different if we add or subtract
the same quantity from both. If a≠b, then for any number c
a+c≠b+c
and
a−c≠b−c
We cannot, in general, add two “not equal” statements and get another “not equal” statement.
Consider 3 ≠ 4 and 8 ≠ 7. These are both true statements, but when we add them (left-to-left
and right-to-right), we get 3 + 8 ≠ 4 + 7. But they are equal!
We can multiply both sides of a “not equal” statement by the same nonzero quantity and
get another true statement. If the quantity is 0, then we end up with 0 ≠ 0, which is false.
We can divide both sides of a “not equal” statement by the same nonzero quantity and
get another true statement. If the quantity by which we divide through is 0, we get undefined
results on both sides of the inequality symbol.
Inequality Morphing 183
