Algebra Know-It-ALL

We can divide both sides of a “larger than or equal” statement by the same positive quantity
and get another valid statement. If a≥b, then for any positive number p

a/p≥b/p

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. If a≥b, then for any negative number n

a/n≤b/n

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

• Can we reverse the order? Not in general, unless we change the inequality to “smaller
  than or equal.”


• 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 nonnegative.


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

ManipulatingÄ statements

If some quantity a is smaller than or equal to another quantity b, we cannot reverse the order,
except when a happens to equal b. It’s not generally true that if a≤b, thenb≤a. But we can
reverse the order if we also reverse the sense of the inequality. If a≤b, then it is always true
thatb≥a.
We can add or subtract the same quantity from each side of a “smaller than or equal”
statement. If a≤b, then for any number c

a+c≤b+c

and

a−c≤b−c

We can always add two “smaller than or equal” statements (left-to-left and right-to-right) and
get another valid 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 “smaller than or equal” statement by the same nonnegative
quantity and get another valid statement. If a≤b, then for any nonnegative number q

aq≤bq

Inequality Morphing 187