Algebra Know-It-ALL

or

− 7 =− 7 =− 7

3. If we multiply an equation through by the number 0, we will always get a statement
   to the effect that 0 equals itself. That’s true, but it’s trivial and isn’t good for much of
   anything. We might multiply an equation through by a variable or expression that’s
   equal to 0, even though we aren’t aware of it at the time. That’s likely to make the
   equation more complicated, but it won’t make it false. Consider this:

x= 2

Let’s multiply this through by (2 −x). We get

x(2−x)= 2(2 −x)

which expands to

2 x−x^2 = 4 − 2 x

In this case, our manipulation does us no good. But it does no real harm either, as the

inadvertent division by 0 can do. Occasionally, a manipulation like this can put a compli-

cated equation into a form that’s easier to work with.

4. Let’s call the set of negative integers Z−. Remember the standard name for the set of
   natural numbers; it’s N. We have

Z−= {..., −5,−4,−3,−2,−1}

and

N= {0, 1, 2, 3, 4, 5, ...}

From these statements, we can see that any negative integer we choose will be smaller than

any natural number we choose. Therefore, if x is an element of Z− and y is an element of N,

x is smaller than y. In logical form along with set notation, we can write this as

[(x∈Z−) & (y∈N)]⇒x < y

5. Let’s call the set of nonpositive reals R 0 − (for “0 and all the negative reals”) and the set
   of nonnegative reals R 0 + (for “0 and all the positive reals”). Both of these sets include
   0, but that’s the only element they share. Therefore, any nonpositive real number we
   choose must be smaller than or equal to any nonnegative real number we choose. In
   other words, if x is an element of R 0 − and y is an element of R 0 +, then x is smaller than
   or equal to y. In logical form along with set notation, we can write this as

[(x∈R 0 −) & (y∈R 0 +)] ⇒x≤y

620 Worked-Out Solutions to Exercises: Chapters 11 to 19