Algebra Know-It-ALL

If the quantity by which we divide through is 0, we get undefined results on both sides of the
inequality symbol. If the quantity by which we divide through is negative, the sense of the
inequality is reversed, just as with multiplication by a negative. If a>b, then for any negative
numbern

a/n<b/n

Here’s a summary of how we can morph “strictly larger than” statements.

• Can we reverse the order? Never, unless we change the inequality to “strictly smaller
  than.”


• Can we add the same quantity to both sides? Yes.


• Can we subtract the same quantity from both sides? Yes.


• Can we add one statement to another? Yes.


• Can we multiply both sides by the same quantity? Only if that quantity is positive.


• Can we divide both sides by the same quantity? Only if that quantity is positive.

Manipulating< statements

If some quantity a is strictly smaller than another quantity b, we cannot reverse the order and
still have a valid statement. It is never true that if a<b, then b<a. But we can reverse the
order if we also reverse the sense of the inequality. If a<b, then it is always true that b>a.
We can add or subtract the same quantity from each side of a “strictly smaller than” state-
ment. If a<b, then for any number c

a+c<b+c

and

a−c<b−c

We can add two “strictly smaller than” statements (left-to-left and right-to-right) and get
another “strictly smaller than” statement. For any numbers a, b, c, and d, if a<b and c<d,
then

a+c<b+d

We can multiply both sides of a “strictly smaller than” statement by the same positive quantity
and get another valid statement. If a<b, then for any positive number p

ap<bp

If the quantity by which we multiply through is 0, then we end up with 0 < 0, which is false. If
the quantity by which we multiply the statement through is negative, the sense of the inequal-
ity is reversed. If a<b, then for any negative number n,

an>bn

Inequality Morphing 185