###### Example3.5.3

Solve for $$x$$ in $$2x+1=2x+1\text{.}$$

We will move all terms containing $$x$$ to one side of the equals sign:

\begin{align*} 2x+1\amp=2x+1\\ 2x+1\subtractright{2x}\amp=2x+1\subtractright{2x}\\ 1\amp=1 \end{align*}

At this point, $$x$$ is no longer contained in the equation. What value can we substitute into $$x$$ to make $$1=1$$ true? Any number! This means that all real numbers are possible solutions to the equation $$2x+1=2x+1\text{.}$$ We say this equation's solution set contains all real numbers. We can write this set using set-builder notation as $$\{x\mid x\text{ is a real number}\}$$ or using interval notation as $$(-\infty,\infty)\text{.}$$

The equation $$1=1$$ is known as an identity as it is always true.

in-context