91Ó°ÊÓ

For any real numbers $a,b$, where $bâ‰\yen 0$, if $\left|a\right|=b$then $a=b$or $a=-b$.

Consider the equation $|x+7|=3x-5$.

Use the concept of absolute value equation as follows:

$x+7=-\left(3x-5\right)$ or $x+7=3x-5$.

Case 1. Simplify $x+7=-\left(3x-5\right)$.

Simplify the equation as follows:

$\begin{array}{l}x+7=−\left(3x−5\right)\\ x+7=−3x+5\\ x+3x=5−7\\ 4x=−2\\ x=\frac{−2}{4}\\ x=−\frac{1}{2}\end{array}$

Since the solution is negative but absolute value cannot be negative therefore no solution for the equation $x+7=-\left(3x-5\right)$.

Case 2. Simplify $x+7=3x-5$.

Simplify the equation as follows:

$\begin{array}{l}x+7=3x−5\\ x−3x=−5−7\\ −2x=−12\\ 2x=12\\ x=\frac{12}{2}\\ x=6\end{array}$

Therefore, the solution is $x=6$or no solution.

Case 2. Substitute $x=6$in the equation $x+7=3x-5$and simplify as follows:

$\begin{array}{l}|x+7|=3x−5\\ |6+7|=3\left(6\right)−5\\ \left|13\right|=18−5\\ 13=13\end{array}$

Since the left-hand side of the equation is equal to the right-hand side of the equation in this case therefore, the solution is $x=6$.