If, when you solve an equation, you get a false statement such as 2 = 3, then the equation has **no solution.**

If, when you solve an equation, you get a single solution such as x = -3, then the equation has **one solution**.

If, when you solve an equation, you get something such as 2 = 2, then the equation has an infinite number of solutions and is in fact an **identity**.

**Example 1**

Solve 6(x + 4) = 6x + 20

**Answer**

6(x + 4) = 6x + 20

6x + 24 = 6x + 20

24 = 20

False so no solution to this equation.

**Example 2**

Solve 6(x + 4) = 6x + 24

**Answer**

6(x + 4) = 6x + 24

6x + 24 = 6x + 24

24 = 24

True for all values of x, so this is an identity.

We say 6(x + 4) ≡ 6x + 4

**Example 3**

Solve 6(x + 4) = 5x + 20

**Answer**

6(x + 4) = 5x + 20

6x + 24 = 5x + 20

x = -4

One solution to this equation.