Algebra Know-It-ALL

186 Equations and Inequalities

We can divide 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

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 “strictly smaller than” statements.

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

Manipulatingê statements

If some quantity a is larger than or equal to another quantity b, we cannot reverse the order,

except in the situation where a happens to equal b. For that reason, it’s not generally true that

ifa≥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 “larger than or equal” state-

ment. If a≥b, then for any number c

a+c≥b+c

and

a−c≥b−c

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

get another valid statement. For any numbers a, b, c, andd, if a≥b and c≥d, then

a+c≥b+d

We can multiply both sides of a “larger 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

If q= 0, 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