###### Example3.6.2

Solve for $$x$$ in $$3x=3x+4\text{.}$$

To solve this equation, we need to move all terms containing $$x$$ to one side of the equals sign:

\begin{align*} 3x\amp=3x+4\\ 3x\subtractright{3x}\amp=3x+4\subtractright{3x}\\ 0\amp=4 \end{align*}

Notice that $$x$$ is no longer present in the equation. What value can we substitute into $$x$$ to make $$0=4$$ true? Nothing! We say this equation has no solution. Or, the equation has an empty solution set. We can write this as $$\emptyset\text{,}$$ which is the symbol for the empty set.

The equation $$0=4$$ is known a false statement since it is false no matter what $$x$$ is. It indicates there is no solution to the original equation.

in-context