Understanding solutions

For purposes of this course, we can consider an equation (or inequality or compound statement) with one variable solved if it is one of the following:

An equation with the variable alone on the left and a (defined, simplified) constant on the right;
An order inequality (not using ‘≠’) with the variable alone on the left and a constant on the right;
A compound inequality using only ‘<’ and ‘≤’ with the variable alone in the middle, a constant on the left, and a larger constant on the right;
Any combination of above statements joined by ‘or’, in increasing order, with no overlapping solutions;
The statement ‘True’;
The statement ‘False’;

Not every Algebra problem can be put in one of these forms; aside from the possibility of an equation with two or more variables (which I'll discuss later), if you take Intermediate Algebra or (worse) Trigonometry, you'll run across equations in only one variable whose solution sets are still more complicated. However, all the equations and inequalities in one variable that we'll study in this course (and many that you'll come across later too) will have solutions in the above forms.

I have put more detail in the specifications here than you really need. For example, it doesn't really matter if the variable is on the left or the right, and you don't always have to give a compound statement in increasing order. But the forms as I've described them are the most common.

Graphing in one variable

To understand these forms better, I'll look at some examples of each. Furthermore, I'll show how each of these examples can be graphed, that is how you can draw a picture of the solutions on a number line. Drawing such a picture (or graph) is not necessary in order to understand a solution, but it's often helpful (especially for inequalities).

To begin with, consider the equation

x = 4.

It's easy to draw a picture of this on a number line:

This is so easy that there may not seem much point to it. However, there are a few points worth mentioning:

The line has arrows at either end, because it goes on forever, even though the solution is at only one place.
A scale is marked on the line aside from just the point 4, so you can see the solution in context.
The line is labelled with the variable x, so you know that the graph is for x = 4, rather than y = 4 or t = 4.

Now consider a more complicated solution, the compound statement

x = −2 or x = 4 or x = 6.

Now the graph looks a little more interesting:

You can see here what I mean when I say that the parts of a compound statement should be listed in increasing order;

x = 4 or x = 6 or x = −2,

means the same thing, but it doesn't match the graph as nicely.

Now consider the inequality

y < 3.

Here, y could be any real number less than 3, so the graph is spread out along that entire part of the line:

Since y := 3 is not itself a solution, I indicate this with a round parenthesis. The parenthesis also faces towards the solutions, which helps if it's hard to see the shading.

Compare this with

y ≥ 3:

Now I have a square bracket instead of a round parenthesis, to indicate that now y := 3 is a solution. Also, the bracket faces the other way, because now the solutions include values larger than 3.

There is another way to graph inequalities, which you may have already learned before this course. I think that it's less clear than the method above, but you can use it if you like. In this method, you use a solid dot (instead of a square bracket) to indicate that a boundary point is a solution, and you use a hollow circle (instead of a round parenthesis) to indicate that a boundary point is not a solution. Then you get these graphs:

If you use the circle/dot method, then be sure to shade in well the region where the solutions are; this isn't so important when you use the parenthesis/bracket method.

Now consider the compound inequality

−3 < x ≤ 2.

This means that −3 < x and x ≤ 2. In other words, x is between −3 and 2, and it can't be exactly −3, but it might be exactly 2. So I shade in the region between −3 and 2, put a round parenentheses (or hollow circle) at −3, and put a square bracket (or solid dot) at 2:

You could just as well write this compound inequality as 2 ≥ x > −3, but again we prefer to write things in increasing order.

Finally, consider the compound statement

x ≤ −3 or x > 2.

This comes in two pieces, one where x ≤ −3, and another where x > 2, so I simply mark these two pieces separately on the same graph:

Again, you could just as well write x > 2 or x ≤ −3, but once more we prefer to write things in increasing order.

Notice that these two pieces have no overlap. Compare this with a statement like

x ≤ 3 or x > −2.

This statement is always true, one way or another, so I should really simplify it further to the simple statement

True.

That has a very simple graph; everything is filled in:

The flip side of this idea is a compound inequality like

2 < x ≤ −3.

Because 2 > −3, no real number can possibly be both greater than 2 and less than −3, so this statement is simply

False.

Its graph is the simplest of all, completely empty:

Solution sets

The solution set of a statement about real numbers is that part of the real line where the statement's solutions are. The graphs above are pictures of such solution sets. But it's also helpful to have an algebraic notation for the solution set. Here are the notations for the solution sets that appear in this course:

If there is just one solution, then put that solution in curly braces.
If there are a few solutions (but only a finite list of them), then put that list in curly braces, separated by commas. (You can list them in any order, but increasing order is usually easiest to understand.)
If there is a limited range of values, as we have for a compound inequality, give the first value and the last value, separated by a comma; put round parentheses or square brackets around this pair, depending on whether these values are or are not included.
If there is an unlimited range of values in the negative direction, then use the symbol ‘−∞’ (pronounced ‘minus infinity’) for the first value; use a round parenthesis there (since −∞ itself is not a real number, so it can't possibly be a solution).
If there is an unlimited range of values in the positive direction, then use the symbol ‘∞’ (pronounced ‘infinity’) for the last value; again use a round parenthesis there.
If there are two or more disjoint ranges of values, then list them all, separated by the symbol ‘∪’ (pronounced ‘union’); again, it's helpful to list these in increasing order.
If every real number is a solution, then the solution set is (−∞, ∞).
If no real number is a solution, then the solution set is ∅, called the empty set.

Again, these rules will probably be clearer with some examples. I'll use the same examples as before; they fit into this table:

Statement:	Solution set
x = 4;	{4};
x = −2 or x = 4 or x = 6;	{−2, 4, 6};
y < 3;	(−∞, 3);
y ≥ 3;	[3, ∞);
−3 < x ≤ 2;	(−3, 2];
x ≤ −3 or x > 2;	(−∞, −3] ∪ (2, ∞);
True;	(−∞, ∞);
False;	∅;

Strictly speaking, you should say that {4} is the solution set for x of the equation x = 4. Similarly, (−3, 2] is the solution set for y of the double inequality −3 < y ≤ 2.

These continuous ranges (as I've been calling them) are intervals in the real line; writing solutions sets this way is called interval notation. (The first couple of examples aren't really interval notation.)

Compare the interval notation to the graphs; you'll see that (except for −∞ and ∞) the round parentheses and square brackets match up perfectly; they're used in the same places, facing in the same directions. This is no coincidence, of course; the notation is designed to work this way!

Usually, it's best to write things as in the left column of the table above. This is more concrete, and it makes clear what variable you're discussing. You usually want the solution sets only in more abstract settings.

Checking solutions

You already know how to check whether a certain assignment of variables is a solution to an equation (or inequality). But you've also been learning techniques for solving equations (and you'll learn more, and in more depth, over the next couple of weeks), so you might wonder why you ever need to check. Just solve the equation; then you'll know what the solutions are!

Actually, there are several good reasons to check solutions:

Some equation-solving techniques do not yield equivalent equations! These techniques sometimes yield extraneous solutions, which appear to be solutions but are not really, so you have to check all of your solutions when you use these techniques. You won't have to deal with these until next term, but it's worth keeping in mind.
Sometimes the equation that you set up for a word problem doesn't take account of all of the relevant information in a problem. (For example, a length should usually be positive, but you might not notice this when you decide to call it l, which in principle might stand for a negative number.) You should always check your answers to word problems to ensure that they make sense in the original context.
If you're only interested in one or a few possible solutions, then probably it's easier simply to check them than to solve the equation from scratch.
If you have a multiple-choice exam (not in this course, but perhaps elsewhere), then it might be a lot easier than you think— if you only have to check the offered solutions. (A ‘none of the above’ option can sometimes cause problems, however.)
If you're not sure whether you solved an equation correctly (do you ever make mistakes?), then checking your solutions is a quick way to verify your answer.

The best advice that I can give for checking solutions is this: Always check your solutions in the original problem! If you only check in a step halfway through, then you'll never catch any mistakes in the first half of your solutions. On the other hand, if you do know that you made a mistake but you're not sure where, then it can help to check your solution in various intermediate steps until you find the first place that it seems to work; the mistake must have come just before that step.

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