# #a<b<c<d#. How do you find the solution(s) of |x-a|+|x-c|=|x-b|+|x-d|? It is verifiable that, for #(a, b, c, d)=(1, 2, 3, 5), x=7/2# is a solution.

##### 2 Answers

You need case analysis.

#### Explanation:

There are several cases, in the point view of the value of

[Case1]

If

This result is inconsistent with the fact

[Case2]

If

For example, if

But this is an inappropriate solution as the result is inconsistent with

Then, proceed to the following cases.

[Case3]

If

[Case4]

When

[Case5]

In this case, there is no solution. The reason is same as [Case1].

Here is the alternative way(drawing the graph)

#### Explanation:

Let

This can be written as a piecewise function:

You can write

Then, draw the two graphs:

The graphs below is for

graph{(abs(x-1)+abs(x-3)-y)(abs(x-2)+abs(x-5)-y)=0 [-6.05, 13.95, -1.64, 8.36]}