Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 28

Solve the equation $4(x - 2) + 2(x + 3) = 6$, and check your solution. If applicable, tell whether the equation is an identity or contradiction.


$
\begin{equation}
\begin{aligned}

4(x - 2) + 2(x + 3) =& 6
&& \text{Given equation}
\\
4x - 8 + 2x + 6 =& 6
&& \text{Distributive property}
\\
6x - 2 =& 6
&& \text{Combine like terms}
\\
6x =& 6 + 2
&& \text{Add $2$ from each side}
\\
6x =& 8
&& \text{Combine like terms}
\\
\frac{6x}{6} =& \frac{8}{6}
&& \text{Divide both sides by $6$}
\\
x =& \frac{4}{3}
&& \text{Reduce to lowest term}



\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

4 \left( \frac{4}{3} - 2 \right) + 2 \left( \frac{4}{3} + 3 \right) =& 6
&& \text{Substitute } x = \frac{4}{3}
\\
\\
4 \left( - \frac{2}{3} \right) + 2 \left( \frac{13}{3} \right) =& 6
&& \text{Work inside parentheses first}
\\
\\
\frac{-8}{3} + \frac{26}{3} =& 6
&& \text{Multiply}
\\
\\
\frac{18}{3} =& 6
&& \text{Add}
\\
\\
6 =& 6
&& \text{True}

\end{aligned}
\end{equation}
$