1.) how do you know if an equation has no real solution (ie, it is a contradiction)?

2.) how do you know if an equation has an infinite number of solutions (ie, it is an identity)?

1) The equation reduces to ...

  (something not zero) = 0

2) The equation reduces to ...

  0 = 0

Step-by-step explanation:

1.) "An equation [that] has no real solution" covers a lot of territory. We assume you're primarily interested in polynomial equations of low degree. For example, consider the equation ...

  x + 1 = x

If we subtract x from both sides of the equation, we get ...

  1 = 0 . . . . false

This is clearly false, so the original equation represents a contradiction.

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Equations like ...

  x^2 = -3

can be rearranged to

  x^2 +3 = 0

Again, the stuff on the left cannot be zero, so this equation has no real solution. (x^2 is non-negative for real values of x.)

2.) When an equation is an identity, the expressions on each side of the equal sign are the same value for all values of the variable(s). Thus, subtracting one of those expressions from the equation gives 0 = 0. For example, consider the equation ...

  x + 2x = 3x

If we were to subtract 3x from both sides of this equation and simplify it, we would get ...

  x + 2x -3x = 0 . . . . 3x subtracted

  x(1+2-3) = 0 . . . . . . combine terms

  0x = 0 . . . . . . . . . . simplify

  0 = 0 . . . . . . true for any value of x, an infinite number of solutions