Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 22

Solve the equation $5(x + 3) + 4x - 5 = 4 - 2x$, and check your solution. If applicable, tell whether the equation is an identity or contradiction.


$
\begin{equation}
\begin{aligned}

5(x + 3) + 4x - 5 =& 4 - 2x
&& \text{Given equation}
\\
5x + 15 + 4x - 5 =& 4 - 2x
&& \text{Distributive property}
\\
9x + 10 =& 4 - 2x
&& \text{Combine like terms}
\\
9x + 2x =& 4 - 10
&& \text{Add $(2x-10)$ from each side}
\\
11x =& -6
&& \text{Combine like terms}
\\
\frac{11x}{11} =& \frac{-6}{11}
&& \text{Divide both sides by $11$}
\\
x =& \frac{-6}{11}
&&

\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

5 \left( \frac{-6}{11} + 3 \right) + 4 \left( \frac{-6}{11} \right) - 5 =& 4 - 2 \left( \frac{-6}{11} \right)
&& \text{Substitute } x = \frac{-6}{11}
\\
\\
5 \left( \frac{27}{11} \right) + 4 \left( \frac{-6}{11} \right) - 5 =& 4 - 2 \left( \frac{-6}{11} \right)
&& \text{Add inside the parentheses}
\\
\\
\frac{135}{11} - \frac{24}{11} - 5 =& 4 + \frac{12}{11}
&& \text{Multiply}
\\
\\
\frac{56}{11} =& \frac{56}{11}
&& \text{True}

\end{aligned}
\end{equation}
$