Algebra Know-It-ALL

You must be careful if you multiply or divide an inequality through by a variable, or by any expression
containing a variable. Suppose you multiply the original inequality in this section through by x. Then
you get

3 x< 7 x

This changes the situation completely! You no longer have a plain statement of fact. Now you have some-
thing that contains an unknown. Keep in mind that x might be positive or negative, or even equal to 0. If
x happens to be negative, the sense of the inequality must be reversed if the statement is to remain true. If
x happens to be 0, the statement becomes false no matter what. It’s best to stick with plain numbers, also
called constants, whenever you multiply or divide an inequality through.

Are you still confused?

A whimsical way to state the above warning is to invoke a time-worn proverb known as Murphy’s law: “If
something can go wrong, it will.” Whenever you do anything to an equation or inequality, ask yourself, “Is
this action completely safe? Can anything bad happen?” If you suspect possible trouble, don’t ignore that
uneasy feeling. Check out the rules in this chapter to be sure you’re “obeying the law”! A single blunder
can cause an error that may remain hidden for some time. But eventually, that error will come around
and bite you.

Here’s a challenge!

Suppose we are given the following inequality, and we are told to derive a statement that clearly indicates
all the real numbers x for which it is true:

x+ 10 < 2 x− 24

In other words, we must “solve the inequality.” How can we do it?

Solution

We can use the rules for inequality morphing to change this statement into something with x on one side
and a numeral on the other. First, let’s add 24 to each side, getting

x+ 10 + 24 < 2 x− 24 + 24

This simplifies to

x+ 34 < 2 x

Now let’s subtract x from each side. That gives us

34 < 2 x−x

which simplifies to

34 <x

Inequality Morphing 189