Section 4.5 Going Deeper: Inequalities
In this section, we talked about the difference between equations and expressions. But there's another type of mathematical statement that is like an equation, but instead of declaring that two quantities represent the same value, we want one to be greater than or less than the other. These statements are known as inequalities.
Definition 4.5.1. Inequality.
A mathematical inequality is a statement of the form \(A \gt B\text{,}\) \(A \lt B\text{,}\) \(A \geq B\text{,}\) or \(A \leq B\text{,}\) where $A$ and $B$ are mathematical expressions. Each statement has an associated interpretation:
\(A \gt B\) means that \(A\) represents a quantity that is greater than the quantity that \(B\) represents.
\(A \lt B\) means that \(A\) represents a quantity that is less than the quantity that \(B\) represents.
\(A \geq B\) means that \(A\) represents a quantity that is greater than or equal to the quantity that \(B\) represents.
\(A \leq B\) means that \(A\) represents a quantity that is less than or equal to the quantity that \(B\) represents.
We often call the first two symbols strict inequalities, emphasizing that we are interested in the more stringent condition. Interestingly, we don't have a formal name for the other two. You will sometimes see them called non-strict, inclusive, or weak inequalities, but there's no consensus term among mathematicians.
Manipulating inequalities is almost identical manipulating equations, though there is one very important distinction. When multiplying or dividing, the axioms only allow for doing this with positive values. You should compare this definition with Definition 1.4.1.
Definition 4.5.2. Axioms of Inequality.
Let \(a\text{,}\) \(b\text{,}\) and \(c\) be real numbers. The axioms of inequality state that
If \(a \gt b\text{,}\) then \(b \lt a\text{.}\)
If \(a \gt b\text{,}\) then \(a + c \gt b + c\text{.}\)
If \(a \lt b\text{,}\) then \(a - c \lt b - c\text{.}\)
If \(a \gt b\) and \(c \gt 0\text{,}\) then \(ac \gt bc\text{.}\)
If \(a \gt b\) and \(c \gt 0\text{,}\) then \(\frac{a}{c} \gt \frac{b}{c}\text{.}\)
You might remember from your previous experiences that when you multiply or divide by a negative number, that you're supposed to flip the inequality. But that's not listed here! What's happening is that this property is not an axiom. We can actually show that this property is a logical consequence of the listed properties. We will show how the negative sign appears in a simple calculation and let you try to prove the general property on your own.
To prove the general case, note that if \(c \lt 0\text{,}\) then \(-c \gt 0\) so that you can multiply both sides by \(-c\) in the first step using the axioms of inequality. From there, you'll need to make a similar manipulation to the one above.
We can expand our definition of what it means to solve an equation so that it applies to inequalities. You should compare this definition with Definition 4.1.4.
Definition 4.5.3. Solve and Solution of an Inequality.
To solve an inequality means to find the value (or values) of the variable (or variables) that make the inequality true. A solution of an inequality is a specific value of the variable (or specific values of the variables) that make the inequality true.
With the exception of worrying about the direction of the inequality, solving inequalities is identical to solving equations. In fact, if you go back to any of the problems from previous sections and replace the equal sign with any of the inequality symbols, the algebraic steps will be identical and you would only need to check whether any of the steps involved multiplying or dividing by a negative number. For example, here is "Try It" exercise \#2, except using instead of in the original problem.
Notice that the change happened right at the step where we divided by -2. It doesn't happen before that step, and it doesn't happen on its own line after that step. It also does not happen because the 9 is negative. It happens because the algebraic manipulation called for dividing by a negative number. Furthermore, if you use the improper presentation from the first section of the book, inequalities lead to some rather unusual and nonsensical mathematical statements. We put such a heavy emphasis on presentation in order to avoid many of the errors that can arise from these things.
A key distinction is that solutions to inequalities are usually not a single value, but (usually) an infinite collection of values. For example, the inequality \(x > 0\) is true when \(x = 1, 2, 3, \ldots\text{.}\) But we have to remember that we're thinking about all possible values, so we have to include decimals such as \(0.1, 0.01, 0.001, 0.0001, \ldots\text{.}\)
The second collection of values hints at a very unusual feature about numbers. There is no smallest solution to the inequality \(x > 0\text{.}\) Another way of saying this is that there is no smallest positive number. For any positive number you can think of, there's always another that's less than your number but still positive. If you've never really thought about why this happens, it's worth taking some time to think about it.
It turns out that this simple notion is connected to a number of profoundly interesting mathematical ideas. Here are just a few questions that this one idea leads to:
There are infinitely many numbers between 0 and 1 and infinitely many numbers between 0 and 0.00000001. Since the first gap is bigger than the second gap, does it make sense to say that the infinity of values in the first interval is larger than the infinity of values in the second? (It turns out that these two infinities are the same size, but that isn't the same as saying that all infinities are the same!)
What happens to the value of \(\frac{1}{x}\) as we let \(x\) take the values \(0.1, 0.01, 0.001, 0.0001, \ldots\text{?}\) It looks like it's getting larger and larger. Does it make sense to say that \(\frac{1}{0} = \infty\text{?}\) (It turns out that it doesn't, but the exact reason this idea fails is a bit subtle.)
Does it make sense to have a number where there are infinite number of zeros before the 1? (It turns out that such a mathematical object can make sense if you think about it the right way, but it's no longer what we would call a number. Instead, it's what me might call a surreal number. And no, this is not a joke.)
We won't go into further detail about any of these topics here, as it goes way beyond the scope of this course. But it is interesting to reflect on how some very basic questions can lead to very profound mathematical ideas.