Chapter 1, Problem 17CR

To determine

To solve: $\left|x-2\right|=\left|4+x\right|$

Answer to Problem 17CR

The solution of the equation is $x=-1$ .

Explanation of Solution

Given information: $\left|x-2\right|=\left|4+x\right|$

Calculation:

  $\left|x-2\right|=\left|4+x\right|$

By equating the expression $x-2$ and the opposite of $4+x$ ,

  $\begin{array}{l}x-2=-(4+x)\\ \Rightarrow x-2=-4-x\\ \Rightarrow x-(-x)=-4-(-2)\\ \Rightarrow x+x=-4+2\\ \Rightarrow 2x=-2\\ \Rightarrow x=-\frac{2}{2}\\ \Rightarrow x=-1\end{array}$

However, by equating the expression $x-2$ and $4+x$ ,

  $\begin{array}{l}x-2=4+x\\ \Rightarrow x-x=4+2\\ \Rightarrow 0=6\end{array}$

This is a false statement. So the original equation has only one solution $x=-1$ .

To check, putting $x=-1$ in the original equation,

Left Hand Side $=\left|x-2\right|$

  $\begin{array}{l}=\left|-1-2\right|\\ =\left|-3\right|\\ =3\end{array}$

$=$ Right Hand Side

Thus, the solution of the equation is $x=-1$ .