- 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’;

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.

To begin with, consider the equation

It's easy to draw a picture of this on a number line:x= 4.

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

Now the graph looks a little more interesting:x= −2 orx= 4 orx= 6.

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

means the same thing, but it doesn't match the graph as nicely.x= 4 orx= 6 orx= −2,

Now consider the inequality

Here,y< 3.

Since

Compare this with

y≥ 3:

Now I have a square bracket instead of a round parenthesis, to indicate that now

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 <This means that −3 <x≤ 2.

You could just as well write this compound inequality as 2 ≥

Finally, consider the compound statement

This comes in two pieces, one wherex≤ −3 orx> 2.

Again, you could just as well write

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

This statement isx≤ 3 orx> −2.

True.That has a very simple graph;

The flip side of this idea is a compound inequality like

2 <Because 2 > −3, no real number can possibly be both greater than 2x≤ −3.

False.Its graph is the simplest of all, completely empty:

- 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

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.

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

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