Algebra Know-It-ALL

190 Equations and Inequalities

We can say this in a more intuitive way by turning it around and reversing the sense of the inequality, so

we have

x> 34

This is the standard way to state the solution to any single-variable algebra problem. We put the variable

all by itself on the left-hand side of the relation symbol, and a plain numeral all by itself on the right.

Here’s a final challenge!

In terms of an inequality statement and set notation, describe how the nonnegative integers relate to the

negative real numbers.

Solution

Let’s call the set of nonnegative integers Z 0 +, and the set of negative reals R−. Any negative real number we

choose will be smaller than any nonnegative integer we choose. Therefore, if x is an element of Z 0 + and y is

an element of R−, then x is larger than y. In logical form along with set notation, we can write this as

[(x∈Z 0 +) & (y∈R−)]⇒x>y

Practice Exercises

This is an open-book quiz. You may (and should) refer to the text as you solve these problems.

Don’t hurry! You’ll find worked-out answers in App. B. The solutions in the appendix may not

represent the only way a problem can be figured out. If you think you can solve a particular

problem in a quicker or better way than you see there, by all means try it!

1. Suppose we see this equation:
   7/2= 14/4 = 21/6
   How can we simplify this using the rules for equation morphing, so we get a statement
   that says a positive integer is equal to itself?


2. How can we morph the equation in Prob. 1 so we get a statement to the effect that a
   negative integer is equal to itself?


3. What happens if we multiply an equation through by the number 0? What happens if
   we multiply an equation through by a variable or expression that ultimately turns out to
   equal 0, although don’t know it at the time?


4. In terms of an inequality statement and set notation, describe how the negative integers
   relate to the natural numbers. Here’s a hint: Use the same approach as we did in the
   final challenge.


5. In terms of an inequality statement and set notation, describe how the nonpositive real
   numbers relate to the nonnegative real numbers. Here’s a hint: Use the same approach
   as we did in the final challenge.