Chapter 1, Problem 17CR

To determine

To solve: the equation.

Answer to Problem 17CR

The required result is $x=-1$

Explanation of Solution

Given:

  $\left|x-2\right|=\left|4+x\right|$

Calculation:

Given

  $\left|x-2\right|=\left|4+x\right|$

Write two related linear equations

  $x-2=4+xorx-2=-\left(4+x\right)$

Solve equation 1:

  $x-2=4+x$

Subtract both sides by $x$

  $x-x-2=4+x-x$

No solution

  $-2\cancel{=}4$

Solve equation 2:

  $x-2=-\left(4+x\right)$

Distribute negative sign

  $x-2=-4-x$

Add both sides by 2

  $x-2+2=-4+2-x$

Add both sides by x

  $x+x=-2-x+x$

Divide both sides by 2

  $\begin{array}{l}\frac{2x}{2}=\frac{-2}{2}\\ x=-1\end{array}$

Check$\left(x=-1\right)$ :

  $\left|x-2\right|=\left|4+x\right|$

Substitute

  $\left|\left(-1\right)-2\right|=\left|4+\left(-1\right)\right|$

Simplify

  $\left|-3\right|=\left|3\right|$

Get absolute value

  $3=3$

Conclusion:

The required result is $x=-1$