**What is Identities and Contradictions ?**

Look at this equation:

x + 1 = 1 + x

It happens to be true always, no matter what the value of x. (Try it out! What if x is 43?) Every number is a solution. Equations like this are called identities. (Because they are always true, they can also be called laws, or rules).

On the other hand, look at this:

x + 1 = x + 5.

This is a contradiction no matter what the value you choose for x, the equation will be false. The solution set of this equation is the empty set.

How can I recognize an identity or a contradiction?

**Example 1: ** Is this equation an identity, a contradiction, or neither?

x(x + 2) = (x + 1)^{2} - 1

We'll simplify and try to solve it:

x^{2} + 2x = (x^{2} + 2x + 1) -1

x^{2} + 2x = x^{2} + 2x.

This is obviously true for all x, so the equation was an identity.

**Example 2: **Is this equation an identity, a contradiction, or neither?

2(x + 1) = 2x + 1

Let's simplify and try to solve it.

2x + 2 = 2x + 1

2 = 1.

If we get to a point where something is obviously false (like 2 = 1!), we know we have a contradiction.