Algebra Know-It-ALL

184 Equations and Inequalities

Here’s a summary of how we can morph “not equal” statements.

• Can we reverse the order? Yes.


• 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? Not in general.


• Can we multiply both sides by the same quantity? Only if that quantity is not 0.


• Can we divide both sides by the same nonzero quantity? Yes.

Manipulating> statements

If some quantity a is strictly larger 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. However, 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 larger than” state-

ment. If a>b, then for any number c

a+c>b+c

and

a−c>b−c

We can always add two “strictly larger than” statements (left-to-left and right-to-right) and

get another “strictly larger than” statement. For any numbers a, b, c, andd,if we have a>b

andc>d,then

a+c>b+d

We can multiply both sides of a “strictly larger 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 happens to be negative, the sense of

the inequality is reversed. The “strictly larger than” relation turns into a “strictly smaller than”

relation. If a>b, then for any negative number n

an<bn

We can divide both sides of a “strictly larger than” statement by the same positive quantity and

get another valid statement. If a>b, then for any positive number p

a/p>b/p