For example, consider the equation
2*x* − 4 = 3*x* − 9.
If you substitute *x* = 4 (say),
then you get this result:

- 2
*x*− 4 = 3*x*− 9 — original equation; - 2(4) − 4 = 3(4) − 9 —
substitute
*x*= 4; - 8 − 4 = 12 − 9 — work out 2 · 4 = 8 and 3 · 4 = 12;
- 4 = 3 — work out 8 − 4 = 4 and 12 − 9 = 3;
- False — since in fact 4 > 3.

- 2
*x*− 4 = 3*x*− 9 — original equation; - 2(5) − 4 = 3(5) − 9 —
substitute
*x*= 5; - 10 − 4 = 15 − 9 — work out 2 · 5 = 10 and 3 · 5 = 15;
- 6 = 6 — work out 10 − 4 = 6 and 15 − 9 = 6;
- True — since in fact 6 = 6.

For another example, consider the equation *t* = 1/*t*.
Let me try *t* = 2, *t* = 1, and *t* = 0.

*t*= 1/*t*— original equation;- 2 = 1/2 — substitute
*t*= 2; - False — since in fact 2 > 1/2.

*t*= 1/*t*— original equation;- 1 = 1/1 — substitute
*t*= 1; - 1 = 1 — work out 1/1 = 1;
- True — since in fact 1 = 1.

*t*= 1/*t*— original equation;- 0 = 1/0 — substitute
*t*= 0; - Undefined — since 1/0 is undefined.

For the most part,
a meaningless equation is just as good or bad as a false one,
and in fact some mathematicians
would count this equation as false when *t* = 0.

An equation may have just one solution, many solutions, or none at all.
As it turns out, *x* = 5
is the *only* solution
to 2*x* − 4 = 3*x* − 9.
However, *t* = 1
is *not* the only solution to *t* = 1/*t*.
This is because *t* = −1 is *also* a solution:

*t*= 1/*t*— original equation;- −1 = 1/(−1) —
substitute
*t*= −1; - −1 = −1 — work out 1/(−1) = −1;
- True — since in fact −1 = −1.

For an example of an equation with *no* solutions,
consider *x* = *x* + 1;
no matter what real number *x* is,
*x* will in fact always be less than *x* + 1,
so nothing could possibly be a solution to this equation.
For an example of an equation with *many* solutions,
consider the equation *x* = |*x*|;
one solution is *x* = 0,
but in fact any positive number could be used instead.
(On the other hand, no negative number gives a solution.)
Or consider the equation *x* = *x*.
No matter what real number you substitute for *x*,
this statement will obviously be true!
Finally, consider the equation 1/*x* = 1/*x*.
Obviously this statement is true whenever its defined,
but *x* = 0 is not a solution,
because the result is undefined in that case.

So in summary, an equation might have no solutions, one real number as a solution, a few solutions, a whole range of solutions, every real number as a solution, or every real number with one or a few exceptions. Pretty much anything is possible if you pick the right equation!

For example, since *x* = 5 is the only assignment
that will make

2true, and it's also (obviously) the only assignment that will makex− 4 = 3x− 9

true, these two equations are equivalent. Similarly, the equationx= 5

is equivalent to the compound statementt= 1/t

because in each case the only solutions aret= 1 ort= −1,

2is equivalent to the inequalityx− 4 > 3x− 9

because in each case, the solutions are given by assigningx< 5,

The book doesn't introduce a symbol to describe equivalence of statements, but you can use ‘⇔’ if you wish. (This symbol is usually read aloud as ‘if and only if’.) For example, the equivalences above can be symbolised thus:

- 2
*x*− 4 = 3*x*− 9 ⇔*x*= 5; *t*= 1/*t*⇔*t*= 1 or*t*= −1;- 2
*x*− 4 > 3*x*− 9 ⇔*x*< 5.

2In this problem, you dox− 4 = 3x− 9 =x= 5. (WRONG!)

- An equation with the variable alone on the left and a (defined, simplified) constant on the right;
- Any number of such equations joined by ‘or’, with no repetitions, listed in increasing order;
- The statement ‘True’ (which is always true); or
- The statement ‘False’ (which is always 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 an ‘or’ statement in increasing order. But the forms as I've described them are the most common.

2(is equivalent to the equationx+y) = 7x

2(and of course there's nothing special about the 7x+ 2y= 7x

This trick has nothing to do with *equations* as such,
and it works just as well for inequalities
(or any other statement one might make about real numbers).
The big idea for solving equations specifically is this:
*Do the same invertible operation to both sides.*
An **invertible operation**
is any operation on a real number
that is always defined
and that has another (inverse) operation
that will always turn the result back into the number you started with.

For example, you can add 5 to any number,
and you get back where you started if you then subtract 5,
so adding 5 is an invertible operation.
Or you can add *x* to any number,
and you get back where you started if you then subtract *x*,
so adding *x* is an invertible operation.
In fact, you can add any defined expression you like,
and you get back where you started if you then subtract that same expression,
so adding any defined expression is always an invertible operation.

Other invertible operations include subtracting any expression,
multiplying by any expression known to be nonzero,
and dividing by any expression known to be nonzero.
For multiplication and division,
it's important that you know that the expression is nonzero;
usually, this means that you can only multiply or divide by constants.
In general, you can't divide by *x*,
because if *x* is 0, then this operation is not defined.
And you can't multiply by *x*,
because if *x* is 0, then this operation is not invertible.
(However, sometimes you're in a situation
where you know that *x* can't be 0;
then it's OK to multiply or divide by *x*.)

Finally, you can always turn an equation into an equivalent equation by swapping the two sides. This may seem silly, but it's sometimes nice to do.

In summary, here's a list of techniques for solving equations:

- Simplify either side (or both);
- Add the same expression to both sides (and then simplify them);
- Subtract the same expression from both sides (and simplify);
- Multiply both sides by the same nonzero constant (and simplify);
- Divide both sides by the same nonzero constant (and simplify);
- Swap the sides.

I'll discuss later another technique useful when you have an absolute value in an expression. You'll learn some others in Intermediate Algebra. There are always more techniques, some of which are still being discovered.

- If there is just one solution,
then put that solution in curly braces.
For example, if
*x*= 4 is the only solution, then the solution set is {4}. - 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.)
For example,
if
*y*= −2 or*y*= 4 or*y*= 6, then the solution set is {−2, 4, 6}. - If
*every*real number is a solution, then the solution set is the set of all real numbers, written**R**. - If
*no*real number is a solution, then the solution set is {}, called the*empty set*.

Usually, it's best to write an answer as a simple statement; this is more concrete, and it makes clear what variable you're discussing. You usually want 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.

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