Chapter 1.4, Problem 42PPS

To determine

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

Answer to Problem 42PPS

The value of $y$ in the given equation is $y=-3$ .

Explanation of Solution

Given information:

Given equation

  $\left|6y+25\right|=-3y-2$

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,

  $\left|6y+25\right|=-3y-2$

Now, solve the equation as shown,

  $\begin{array}{c}\left|6y+25\right|=-3y-2\\ 6y+25=-3y-2\text{or}6y+25=-\left(-3y-2\right)\\ 6y+25=-3y-2\text{or}6y+25=3y+2\\ 6y+25+3y-25=-3y-2+3y-25\text{or}6y+25-3y-25=3y+2-3y-25\\ 9y=-27\text{or}3y=-23\\ y=-3\text{or}y=-\frac{23}{3}\end{array}$

Now, check the result by back substitution.

  $\begin{array}{c}\left|6y+25\right|=-3y-2\\ \left|6\left(-3\right)+25\right|=-3\left(-3\right)-2\\ \left|-18+25\right|=9-2\\ \left|7\right|=7\\ 7=7\end{array}$

And

  $\begin{array}{c}\left|6y+25\right|=-3y-2\\ \left|6\left(-\frac{23}{3}\right)+25\right|=-3\left(-\frac{23}{3}\right)-2\\ \left|-46+25\right|=21-2\\ \left|21\right|=19\\ 21=19\end{array}$

Which is false.

Hence, the value of $y$ in the given equation is $y=-3$ .