Example 2.5.2

A number plus \(2\) is \(6\text{.}\) What is that number?

You may be so familiar with basic arithmetic that you know the answer already. The algebra approach would be to start by translating “A number plus \(2\) is \(6\)” into a math statement — in this case, an equation:

\begin{equation*} x+2=6 \end{equation*}

where \(x\) is the number we are trying to find. In other words, what should be substituted in for \(x\)…

\begin{equation*} x+2=6 \end{equation*}

… to make the equation true?

Now, how do you determine what \(x\) is? One valid option is to just imagine what number you could put in place of \(x\) that would result in a true equation. Would \(0\) work? No, that would mean \(0+2\stackrel{\text{no}}{=}6\text{.}\) Would \(17\) work? No, that would mean \(17+2\stackrel{\text{no}}{=}6\text{.}\) Would \(4\) work? Yes, because \(4+2=6\) is a true equation.

So one solution to \(x+2=6\) is \(4\text{.}\) No other numbers are going to be solutions, because when you add \(2\) to something smaller or larger than \(4\text{,}\) the result is going to be smaller or larger than \(6\text{.}\)