Q37. - Algebra 2

Question

Solve the equation and check the solution.

$|x+7|=3x-5$

Step-by-Step Solution

Answer

The solution is $x=6$ or no solution.

Step 1- Apply the concept of absolute value.

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

Step 2- Step description.

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

Use the concept of absolute value equation as follows:

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

Step 3- Step description.

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.

Step 4- Verify the 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 $x=6$.