`length`

or `population`

,
or even nonsense symbols like ↵ or ◊.
I'll mostly use ordinary English letters.
So if you want to say that it doesn't matter what order you add two real numbers, then rather than saying 3 + 4 = 4 + 3 (since they're both 7) and 2 + 9 = 9 + 2 (since they're both 11) and so on, you can just say that

for all real numbersa+b=b+a

and see if you can figure out whatx+ 2 = 5

wherel= 2w,

Probably the most important technique to use in algebra
is *substitution*.
If two things are known (or assumed) to be equal,
then you can replace one of them with the other in any expression.
For example, you know that 3 + 7 = 10,
so 3 + 7 + 5 = 10 + 5;
you also know that 10 + 5 = 15, so 3 + 7 + 5 = 15.
Similarly, if (in some problem) you know that *x* = 3,
then *x* + 5 = 3 + 5;
you also know that 3 + 5 = 8, so *x* + 5 = 8.
Finally, if you assume that *x* = 7,
then the inequlity *x* − 2 < 5
means the same thing as the inequality 7 − 2 < 5;
since 7 − 2 = 5 and it is false that 5 < 5,
it must also be false that *x* − 2 < 5.
If you know that *x* − 2 < 5 really,
then your assumption that *x* = 7 must also be false.

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This web page was written in 2007 and 2008 by Toby Bartels, last edited on 2010 October 5. Toby reserves no legal rights to it.

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