Chapter 3.7, Problem 8LC

To determine

To calculate: the similarities and differences of a given equation.

Answer to Problem 8LC

The solutions for the equation $|x-1|=2$ are: $-$ 1 and $3.$

Explanation of Solution

Given information:

The system equations are: absolute value equation $|x-1|=2,$ inequality $|x-1|\le 2$ and $|x-1|\ge 2$ .

Calculation:

For the given absolute value equation $|x-1|=2,$ write the linear pair of equations, $x-1=2$ and $x-1=-2$

Solving each equation separately,

  $\begin{array}{r}\hfill x-1=2x-1=-2\\ \hfill x=3x=-1\end{array}$

The solution to the absolute value equation $|x-1|=2$ are $x=3$ and $x=-1$

  

The inequality $|x-1|\le 2$ can be written as $-2\le x-1\le 2$ .

Solving further we get,

  $-2\le x-1\le 2$

  $-1\le x\le 3$

The solution to the inequality $|x-1|\le 2$ are $-1\le x\le 3$

  

The inequality $|x-1|\ge 2$ can be written as $x-1\le -2$ or $x-1\ge 2$ .

Solving further we get,

  $\begin{array}{l}x-1\le -2x-1\ge 2\\ x\le -1x\ge 3\end{array}$

The solution to the inequality $|x-1|\ge 2$ are $x\le -1$ or $x\ge 3$

  

Observe that there are only 2 equations for the inequalities $|x-1|\le 2$ and $|x-1|\ge 2,$ which are $-$ 1 and 3

Similarly, the solutions for the equation $|x-1|=2$ are $-$ 1 and $3.$

Hence, the solutions for the equation $|x-1|=2$ are $-$ 1 and $3.$