# The inequality | x − 1 | − | x − 3 | ≥ 5 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1, Problem 4P
To determine

## To solve: The inequality |x−1|−|x−3|≥5.

Expert Solution

Solution:

There are no solutions.

### Explanation of Solution

Given:

The equation is |x1||x3|5.

Definition used:

An absolute function is defined as, |x|={x   if x0x if x<0.

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

|x1|={x1       if x10(x1) if x1<0 and |x3|={x3       if x30(x3) if x3<0

Simplify further as, |x1|={x1   if x1x+1 if x<1 and |x3|={x3   if x3x+3 if x<3.

Therefore, there exist three cases such as, x<1,1x<3 and x3.

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

|x1||x3|5x+1(x+3)5x+1+x3525

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

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

|x1||x3|5x1(x+3)5x1+x352x9

The value of x4.5 which is not on the interval, 1x<3.

Case 3: If x3, solve the given inequality as follows.

|x1||x3|5x1(x3)5x1x+3525

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<1,1x<3 and x3.

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