Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 42

Solve the equation $4 [6 - (1 + 2x)] + 10x = 2 (10 - 3x) + 8x$, and check your solution. If applicable, tell whether the equation is an identity or contradiction.


$
\begin{equation}
\begin{aligned}

4 [6 - (1 + 2x)] + 10x =& 2 (10 - 3x) + 8x
&& \text{Given equation}
\\
4[6-1-2x] + 10x =& 20-6x + 8x
&& \text{Distributive property}
\\
4[5-2x] + 10x =& 20 + 2x
&& \text{Combine like terms}
\\
20 - 8x + 10x =& 20 + 2x
&& \text{Distributive property}
\\
2x + 20 =& 2x + 20
&& \text{Combine like terms}
\\
2x - 2x =& 20 - 20
&& \text{Subtract $(2x + 20)$ from each side}
\\
0 =& 0
&&

\end{aligned}
\end{equation}
$


The final line, $0=0$ indicates that the solution set is $\{$ all real numbers $\}$ and the equation $4[6 - (1 + 2x)] + 10x = 2(10 - 3x) + 8x$ is an identity.