Chapter EP, Problem 1.4.25EP

To determine

To solve: The equation $\left|2b+4\right|-3=6b+1$ .

Answer to Problem 1.4.25EP

The solution of the equation $\left|2b+4\right|-3=6b+1$ is 0.

Explanation of Solution

Given:

The equation is $\left|2b+4\right|-3=6b+1$ .

Calculation:

Remove the mod from the equation.

  $\begin{array}{c}\left|2b+4\right|-3+3=6b+1+3\\ \left|2b+4\right|=6b+4\\ 2b+4=\pm \left(6b+4\right)\end{array}$

Simplify the equation $2b+4=\left(6b+4\right)$ .

  $\begin{array}{c}2b+4-4=\left(6b+4\right)-4\\ 2b-6b=0\\ b=0\end{array}$

Simplify the equation $2b+4=-\left(6b+4\right)$ .

  $\begin{array}{c}2b+4-4=-\left(6b+4\right)-4\\ 2b=-6b-8\\ 8b=-8\\ b=-1\end{array}$

Therefore, the solutions of the equation $\left|2b+4\right|-3=6b+1$ are $\left(0,-1\right)$ .

To check the solution, substitute these values in the original equation.

Substitute $b=0$ in the equation $\left|2b+4\right|-3=6b+1$ .

  $\begin{array}{c}\left|2\left(0\right)+4\right|-3=6\left(0\right)+1\\ \left|4\right|-3=1\\ 1=1\\ \text{LHS}=\text{RHS}\end{array}$

Substitute $b=-1$ in the equation $\left|2b+4\right|-3=6b+1$ .

  $\begin{array}{c}\left|2\left(-1\right)+4\right|-3=6\left(-1\right)+1\\ \left|2\right|-3=-6+1\\ -1\ne -5\\ \text{LHS}\ne \text{RHS}\end{array}$

So, the solution $b=0$ is correct.

Therefore, the solution of the equation $\left|2b+4\right|-3=6b+1$ is 0.