Chapter 1, Problem 35SGR

To determine

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

Answer to Problem 35SGR

The value of $a$ in the given equation is $a=10\text{or}a=2$ .

Explanation of Solution

Given information:

Given equation

  $4\left|a-6\right|=16$

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,

  $4\left|a-6\right|=16$

Now, solve the equation as shown,

  $\begin{array}{c}4\left|a-6\right|=16\\ \frac{4\left|a-6\right|}{4}=\frac{16}{4}\\ \left|a-6\right|=4\\ a-6=4\text{or}a-6=-4\\ a-6+6=4+6\text{or}a-6+6=-4+6\\ a=10\text{or}a=2\end{array}$

Now, check the result by back substitution.

  $\begin{array}{c}4\left|a-6\right|=16\\ 4\left|10-6\right|=16\\ 4\left|4\right|=16\\ 16=16\end{array}$

And

  $\begin{array}{c}4\left|2-6\right|=16\\ 4\left|2-6\right|=16\\ 4\left|-4\right|=16\\ 16=16\end{array}$

Which is true.

Hence, the value of $a$ in the given equation is $a=10\text{or}a=2$ .