Intermediate Algebra, Chapter 2, Test, Section Test, Problem 4

Evaluate the given equations below then tell whether the equation is a conditional equation, an identity or a contradiction.
a.) $3x - (2 - x) + 4x + 2 = 8x + 3$
b.) $\displaystyle \frac{x}{3} + 7 = \frac{5x}{6} - 2 - \frac{x}{2} + 9$
c.) $-4(2x - 6) = 5x + 24 - 7x$


$
\begin{equation}
\begin{aligned}
\text{a.) } 3x - 2 + x + 4x + 2 &= 8x + 3
&& \text{Apply Distributive Property}\\
\\
8x &= 8x + 3
&& \text{Combine like terms}\\
\\
0 &\neq 3
\end{aligned}
\end{equation}
$


The system has no solution. Thus, the equation is a contradiction.

$
\begin{equation}
\begin{aligned}
\text{b.) } 2x + 42 &= 5x - 12 - 3x + 54
&& \text{Multiply each side by the LCD } 6 \\
\\
2x + 42 &= 2x + 42
&& \text{Combine like terms}\\
\\
0 &= 0
\end{aligned}
\end{equation}
$


The system has infinitely many solution. Thus, the equation is an identity.


$
\begin{equation}
\begin{aligned}
\text{c.) } -8x + 24 &= 5x + 24 - 7x
&& \text{Apply Distributive Property}\\
\\
-8x + 24 &= -2x + 24
&& \text{Combine like terms}\\
\\
-8x +2x &= 24 - 24
&& \text{Solve for } x\\
\\
-6x &= 0 \\
\\
x &= 0
\end{aligned}
\end{equation}
$


The equation has a solution. Thus, the equation is a conditional.