Algebra Know-It-ALL

188 Equations and Inequalities

If q= 0, then we end up with 0 ≤ 0, which is true. If the quantity by which we multiply the

statement is negative, the sense of the inequality is reversed. If a≤b, then for any negative

numbern

an≥bn

We can divide both sides of a “smaller 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 “smaller than or equal” statements.

• Can we reverse the order? Not in general, unless we change the inequality to “larger
  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.

Are you confused?

Some people find it hard to see why multiplying an inequality through by a negative number reverses its

sense. Here’s an example:

3 < 7

If you multiply this through by −1 without changing the sense of the inequality, you get

3 × (−1)< 7 × (−1)

which simplifies to

− 3 <− 7

This new statement is false! It becomes true if, but only if, you switch the inequality symbol from “strictly

smaller than” to “strictly greater than” getting

− 3 >− 7