Algebra Know-It-ALL

178 Equations and Inequalities

number such as 2. Once you remember this, it’s easy to see that if x is any negative number, then 2x is

smaller than x.Now you can write

If x< 0, then 2x<x

and

If x≤ 0, then 2x≤x

Check these facts out with some actual numbers and you’ll see how they work. When any number is nega-

tive to begin with, doubling it makes it more negative, and therefore smaller.

Logical implication

Here is a new mathematical symbol. An “if/then” statement, such as those above, can be

abbreviated using a double-shafted arrow pointing to the right, often with a little extra space

on either side (⇒), between the “if ” part of the statement and the “then” part. This arrow

stands for the term logically implies, which in plain English translates to “means it is always

true that.” (It does not mean “causes”!) With the help of this symbol, the above facts can be

shortened to

(x< 0) ⇒ (2x<x)

and

(x≤ 0) ⇒ (2x≤x)

Try reading these statements by saying “logically implies” or “means it is always true that”

when you see the arrow.

In any logical implication of this kind, the part of the statement to the left of the arrow

is called the antecedent. The part of the statement to the right of the arrow is called the

consequent.

Here’s a challenge!

Write a pair of “if/then” statements that precisely define all the real numbers that, when divided by 10,

become smaller than the original number.

Solution

Let’s begin by seeking out all the real numbers that become strictly smaller when we divide them by 10.

It’s not difficult to see that any positive real will work. We can say that

(x> 0) ⇒ (x/10<x)

If we start with a negative real and then divide it by 10, the result gets less negative, meaning that it

becomes larger. All the negative reals therefore fail to “qualify.” We want the number to get smaller, not