# How do you solve and graph the inequality #abs(4 – v)< 5#?

##### 2 Answers

Let's start from a graph of a function

For non-negative

For negative

That results in this graph of

graph{|x| [-10, 10, -5, 5]}

Now let's draw a graph of

According to principles of graph transformation, a graph of

graph{|x-4| [-10, 10, -5, 5]}

Now, to solve

Outside of this segment, that is if

For those interested in purely algebraic solution, here is how to do it.

Since, by definition,

we will consider two cases.

Case 1. Seeking solutions that satisfy the inequality

In this case

Solution to this is

Combined with the condition

Case 2. Seeking solutions that satisfy the inequality

In this case

Solution to this is

Combined with the condition

Now it's appropriate to combine two segments that represent the solutions of an original inequality into one segment since these segments are adjacent:

As you see, we get the same solution as using a graph above (which should not be a surprise).