Chapter 1, Problem 4P

To determine

To solve: The inequality $\left|x-1\right|-\left|x-3\right|\ge 5$.

Answer to Problem 4P

Solution:

There are no solutions.

Explanation of Solution

Given:

The equation is $\left|x-1\right|-\left|x-3\right|\ge 5$.

Definition used:

An absolute function is defined as, $\left|x\right|=\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}$.

From the definition, the given equation is expressed as follows.

$\left|x-1\right|=\{\begin{array}{l}x-1\text{ if }x-1\ge 0\\ -\left(x-1\right)\text{ if }x-1<0\end{array}$ and $\left|x-3\right|=\{\begin{array}{l}x-3\text{ if }x-3\ge 0\\ -\left(x-3\right)\text{ if }x-3<0\end{array}$

Simplify further as, $\left|x-1\right|=\{\begin{array}{l}x-1\text{ if }x\ge 1\\ -x+1\text{ if }x<1\end{array}$ and $\left|x-3\right|=\{\begin{array}{l}x-3\text{ if }x\ge 3\\ -x+3\text{ if }x<3\end{array}$.

Therefore, there exist three cases such as, $x<1,1\le x<3\text{ and }x\ge 3$.

Case 1: If $x<1$, solve the given inequality as follows.

$\begin{array}{l}\left|x-1\right|-\left|x-3\right|\ge 5\\ -x+1-\left(-x+3\right)\ge 5\\ -x+1+x-3\ge 5\\ -2\ge 5\end{array}$

The above inequality is false. So, there are no solutions on this interval.

Case 2: If $1\le x<3$, solve the given inequality as follows.

$\begin{array}{l}\left|x-1\right|-\left|x-3\right|\ge 5\\ x-1-\left(-x+3\right)\ge 5\\ x-1+x-3\ge 5\\ 2x\ge 9\end{array}$

The value of $x\ge 4.5$ which is not on the interval, $1\le x<3$.

Case 3: If $x\ge 3$, solve the given inequality as follows.

$\begin{array}{l}\left|x-1\right|-\left|x-3\right|\ge 5\\ x-1-\left(x-3\right)\ge 5\\ x-1-x+3\ge 5\\ 2\ge 5\end{array}$

The above inequality is false. So, there are no solutions on this interval.

Combine the cases 1, 2, and 3, the inequality has no solutions on any of the intervals $x<1,1\le x<3\text{ and }x\ge 3$.