Chapter 1, Problem 37SGR

To determine

To solve: the given equation for $b$ . Check the result.

Answer to Problem 37SGR

The equation has no solution.

Explanation of Solution

Given information:

Given equation

  $\left|b+5\right|=2b-9$

Concept used:

Absolute value of a function a is given by

  $\left|a\right|=\{\begin{array}{l}aa>0\\ -aa<0\\ 0a=0\end{array}$ \

And

  $\begin{array}{l}\left|-a\right|=a\\ \left|a\right|=a\end{array}$

More over

  $\begin{array}{l}\left|x+a\right|=b\\ \Rightarrow x+a=b\text{or}x+a=-b\end{array}$

Calculation:

Consider the given equation,

  $\left|b+5\right|=2b-9$

Now, solve the equation as shown,

  $\begin{array}{c}\left|b+5\right|=2b-9\\ b+5=\left(2b-9\right)\text{or}b+5=-\left(2b-9\right)\\ b+5=2b-9\text{or}b+5=-2b+9\\ b+5-2b=2b-9-2b\text{or}b+5+2b=-2b+9+2b\\ -b+5=-9\text{or}3b+5=9\\ -b+5-5=-9-5\text{or}3b+5-5=9-5\\ -b=14\text{or}3b=4\\ b=-14\text{or}b=\frac{4}{3}\end{array}$

Now, check the result by back substitution.

  $\begin{array}{c}\left|\frac{4}{3}+5\right|=2\left(\frac{4}{3}\right)-9\\ \left|\frac{19}{3}\right|=\frac{8}{3}-9\\ \frac{19}{3}=-\frac{19}{3}\end{array}$

And

  $\begin{array}{c}\left|-14+5\right|=2\left(-14\right)-9\\ \left|-9\right|=-37\\ 9=-37\end{array}$

Which is false.

Hence, the equation has no solution.