Example 3.6.2

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.