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: 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:

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:

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:

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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