Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 38

Solve the equation $- [6x - (4x + 8)] = 9 + (6x + 3)$, and check your solution. If applicable, tell whether the equation is an identity or contradiction.


$
\begin{equation}
\begin{aligned}

- [6x - (4x + 8)] =& 9 + (6x + 3)
&& \text{Given equation}
\\
- [6x - 4x - 8] =& 9 + (6x+3)
&& \text{Distributive property}
\\
- [2x - 8] =& 6x + 12
&& \text{Combine like terms}
\\
-2x + 8 =& 6x + 12
&& \text{Distributive property}
\\
-2x - 6x =& 12-8
&& \text{Subtract $(6x + 8)$ from each side}
\\
-8x =& 4
&& \text{Combine like terms}
\\
\frac{-8x}{-8} =& \frac{4}{-8}
&& \text{Divide both sides by $-3$}
\\
x =& - \frac{1}{2}
&& \text{Reduce to lowest term}

\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

- \left[ 6 \left( - \frac{1}{2} \right) - \left( 4 \left( - \frac{1}{2} \right) + 8 \right) \right] =& 9 + \left( 6 \left( - \frac{1}{2} \right) + 3 \right)
&& \text{Substitute } x = - \frac{1}{2}
\\
\\
- [-3-(-2 + 8)] =& 9 + (-3 + 3)
&& \text{Work inside parentheses first}
\\
\\
-(-3 - 6) =& 9 + 0
&& \text{Simplify}
\\
\\
-(-9) =& 9
&& \text{Simplify}
\\
\\
9 =& 9
&& \text{True}

\end{aligned}
\end{equation}
$