Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 32

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


$
\begin{equation}
\begin{aligned}

-2x - 3(4 - 2x) =& 2(x - 3) + 2
&& \text{Given equation}
\\
-2x - 12 + 6x =& 2x - 6 + 2
&& \text{Distributive property}
\\
4x - 12 =& 2x - 4
&& \text{Combine like terms}
\\
4x - 2x =& -4 + 12
&& \text{Subtract $(2x-12)$ from each side}
\\
2x =& 8
&& \text{Combine like terms}
\\
\frac{2x}{2} =& \frac{8}{2}
&& \text{Divide both sides by $2$}
\\
x =& 4
&&

\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

-2(4) - 3(4-2(4)) =& 2(4-3) + 2
&& \text{Substitute } x = 4
\\
-2(4) - 3(4-8) =& 2(4-3) + 2
&& \text{Multiply inside parentheses first}
\\
-2(4) - 3(-4) =& 2(1) + 2
&& \text{Work inside parentheses first}
\\
-8 + 12 =& 2 + 2
&& \text{Multiply}
\\
4 =& 4
&& \text{True}

\end{aligned}
\end{equation}
$