The first is when we have what is called infinite solutions. This happens when all numbers are solutions. This situation means that there is no one solution. In terms of our treasure hunt, it means that we can't find the treasure because the location of the treasure can be anywhere. There is no x that marks the spot. Our x here marks the whole world, which doesn't help us.
The equation 2x + 3 = x + x + 3 is an example of an equation that has an infinite number of solutions. Let's see what happens when we solve it. We first combine our like terms. We see two x terms that we can combine to make 2x.
2x + 3 = 2x + 3.
Now we can subtract 3 from both sides: 2x = 2x. Hmm. This is an interesting situation; both sides are equal to each other. How many different values of x will make this equation true? Why, isn't it any number? Yes, and so we have our infinite solutions.